Quantifying Galactic Propagation Uncertainty in
WIMP Dark Matter Search with AMSO1 Z=-1
Spectrum
by
Sa Xiao
INSTITTE
OF TECHNOLOGY
SSACHUSETTS
MA
AUG 0 4 2009
LIBRARIES
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
ARCHIVES
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2009
@ Massachusetts Institute of Technology 2009. All rights reserved.
A uthor ..................................
Department of Physics
May 30, 2009
f,n
C ertified by .................................
Peter Fisher
Professor
Thesis Supervisor
Accepted by ..............
," Thomas Greytak
Associate Department Head for Education
Quantifying Galactic Propagation Uncertainty in WIMP
Dark Matter Search with AMS01 Z=-1 Spectrum
by
Sa Xiao
Submitted to the Department of Physics
on May 30, 2009, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
A search for a WIMP dark matter annihilation signal is carried out in the AMS01
negatively charged (Z=-I) particle spectrum, following a set of supersymmetric benchmark scenarios in the mSUGRA framework. The result is consistent with no dark
matter, assuming a smooth isothermal distribution of dark matter in the Galactic
halo. 90% upper bounds of the boost factor by which the flux from the DM annihilation could be enhanced without exceeding AMS01 data are derived to be r
102 - 105,
varied as different mSUGRA senarios. The Boron-to-Carbon ratio energy spectrum
is measured with AMS01, which allows us to constrain the cosmic ray (CR) Galactic
propagation parameters. In the diffusive reaccelaration (DR) model, the propagation
parameters are shown to be D,, N 4.5 x 1028 - 6 x 1028 cm 2 S-1, and VA - 28 - 42
km s-' . The impact of the uncertainties in the cosmic ray propagation model on dark
matter limits is studied and the associated uncertainties of the 90% upper bound of
the boost factor are found to be less than 30%.
Thesis Supervisor: Peter Fisher
Title: Professor
Acknowledgments
I would like to thank a number of people for making this thesis possible and my
graduate experience a life-long treasure: My thesis adviser, Prof. Peter Fisher, for
guiding and helping me through all the ups and downs of my study, research and life.
Without him nothing would be possible. Prof. Sam Ting for giving me this great
opportunity to come to MIT and join the AMS experiment. Prof. Ulrich Becker
for teaching me the essence of experimental technique. Gray Rybka for building the
foundation of this thesis and giving me incredible help throughout my study and
research. James Battat for giving me much valuable advice on my thesis and his
editing help. Gianpaolo Carosi and Ben Monreal for patiently showing me much of
the details of AMS analysis. Feng Zhou for showing me the Earth magnetic field
tracing program and some nice discussion with him. Igor Moskalenko for his help on
GALPROP. Prof. Kate Scholberg for guiding me to my new graduate life at MIT
when I first came.
Thanks should also go to the whole AMS collaboration. Without the successful completion of AMS01 experiment long before I joined, this thesis would not be
possible.
I would also like to thank many people in the neutrino-dark matter group for
their support during my graduate career and their friendship: Prof. Janet Conrad,
Jocelyn Monroe, Prof. Dick Yamamoto, Prof. Joseph Formaggio, Prof. Gabriella
Sciolla, Frank Taylor, Shawn Henderson, Scott Hertel, Asher Kaboth, Richard Ott,
Tom Walker, Lindley Winslow.
Finally I would like to thank my parents for their love and moral support. All my
courage to combat obstacles and pursue dreams comes from their love.
Contents
1 Introduction
9
2 Dark Matter
2.1 Evidence for Dark Matter ...................
.....
2.2 Dark Matter Candidates ...................
......
2.2.1 WIMPs .............
.........
..
2.2.2 Supersymmetric Dark Matter ..................
2.3 (c)MSSM and Benchmark Points . ..................
.
2.4 DM Search Methods ...................
.......
2.4.1 DM Search w/ AMS01 ...................
...
2.5 Substructure of DM Distribution ...................
.
11
11
12
12
13
14
15
16
17
3 Cosmic Rays
3.1 Overview of the Galactic Cosmic Rays .................
3.2 Propagation of the Cosmic Ray Nuclei in the Galaxy . ........
3.2.1 Cosmic Ray Propagation Model . ................
3.2.2 B/C - A Probe of the Propagation Model . .........
3.2.3 Propagation Parameters ...................
3.3 Solar Modulation and Geomagnetic Effects . ..............
19
19
20
20
22
25
25
4 AMS Experiment
4.1 The AMS01 Detector ...................
........
4.1.1 The Magnet ...................
.......
4.1.2 TOF .......
..
......
.............
4.1.3 Tracker ...............
..........
4.1.4 The Aerogel Threshold Cerenkov Counter (ATC) ......
4.1.5 The Anti-Coincidence Counter (ACC) . ............
4.2 The AMS01 Flight ..................
.........
4.2.1
Flight Parameters . ..................
.....
4.2.2 Trigger and Data Acquisition . ...........
. .
5 WIMP annihilation signal search procedure
5.1 AMSO1 Z=-1 Momentum Spectrum . ..................
5.1.1 Data Selection ...................
5.1.2 Exposure Time ...................
......
.......
..
.
.
.
.
27
27
27
27
28
30
30
30
30
31
33
33
33
35
5.2
..........
5.1.3 Acceptance
........
Fit
5.1.4 Background
Neutralino Annihilation Electron and Antiproton Signal Production
6 The B/C Ratio
6.1 Charge Identification ............
6.1.1 Energy Loss Of Heavy Nuclei . . .
6.1.2 Cluster Selection . . . . . . . . ..
6.2 Event Selection ...............
6.2.1 Charge Selection . .........
6.2.2 Cosmic Origin Selection . . . . . .
...................
6.3 Fitting . .
6.4 Combining With Proton Fit . . . . . . ..
6.5 Comparison with Other Measurements . .
6.6 Global Constraints on Propagation Models
41
41
41
42
42
42
47
47
51
54
54
7 Results
7.1 Constraining the Boost Factor . . . . . . .
7.2 Propagation Model Uncertainties . . . . .
59
59
61
8
PAMELA Results
...............
8.1 The p/p Ratio .
..........
Only
8.1.1 Background
8.1.2 Background + DM Signal .....
8.2 The e+ /(e + + e-) ratio ...........
8.2.1 Background Only ..........
.........
8.2.2 Consistency Check
8.2.3 Background + DM signal .....
9 Conclusion
0 1
ha/Iin heiccaro "-f This Analr is
9.2
9.3
9.4
Broaden the Interpretation of the Boost Factor
Message from PAMELA First Results .....
.................
Future Prospects
9.4.1 B/C Ratio Measurement .........
Moving to Positron and Antiproton Channels
9.4.2
A Derivation of Alfven Velocity
74
B GALPROP Parameters (default setting)
77
C Unfolding Method: Convergent Weights
85
D FitMethod
87
Chapter 1
Introduction
Many observations imply the existence of dark matter [21]. However, its nature still
remains mysterious. Weakly interacting massive particles (WIMPs) are promising
candidates [21], particularly the lightest supersymmetric particle (LSP), which may
self-annihilate producing standard model charged particles. If this is the case, we
will have a chance to detect WIMPs indirectly through their annihilation products in
cosmic rays [21].
The AMSO1 (Alpha Magnetic Spectrometer) experiment, a magnetic spectrometer
designed to measure charged cosmic rays from several hundred MeV to about 300
GeV (see chapter 4 see for a description of the AMS experiment), collected data
for ten days while on board the STS-91 Flight of Discovery. Previous work [24]
[55] carried out a search for a WIMP annihilation signal in the AMSO1 negatively
charged (Z=-1) particle spectrum. The result was consistent with no dark matter,
assuming a smooth isothermal distribution of dark matter in the halo, for the set of
supersymmetric benchmark scenarios proposed in [17] in the constrained Minimum
Supersymmetric Standard Model (cMSSM) framework. However, many theories and
numerical simulations predict clumps of dark matter may exist [22] [41]. For clumpy
halos, the rate of WIMP annihilation may be enhanced by orders of magnitude. Such
magnification resulting from the clumpiness of dark matter is parameterized by the
boost factor. We can derive an upper bound of the boost factor by which the electron
flux from the DM annihilation could be enhanced without exceeding AMS01 data.
The modeling of charged particle propagation in the galaxy [58] introduces an
uncertainty into any WIMP search using cosmic rays. In this thesis I study the impact of uncertainties in the cosmic ray propagation model on dark matter limits. The
transport mechanisms of cosmic rays in the galaxy are dominated by particle diffusion in the galactic magnetic fields. Some elements, such as Carbon, Nitrogen and
Oxygen, are directly injected into cosmic rays by supernova explosions. These are
called primary nuclei. Other elements, such as Lithium, Beryllium and Boron, on the
other hand, are mainly generated during cosmic ray propagation, since these light
elements would have been destroyed during nucleosynthesis in the sources. These
secondary nuclei can be used to measure the amount of interstellar matter through
which the primary nuclei pass between the source and the Earth. The ratio of the
number of secondary to the primary elements is largely independent of the injection
spectra, and is a good probe of the propagation model. The Boron-to-Carbon ratio
is a very effective secondary-to-primary ratio for such purpose because they are relatively well measured in the cosmic rays and the cross sections for the production of
their secondaries are well known. AMSO1 had good capability to distinguish charged
particles up to Oxygen (Z=8). The major tasks of this work are: (1) Measure the
Boron-to-Carbon ratio as a function of energy using AMSO1 data, and use it to constrain the galactic propagation parameters, such as the spatial diffusion coefficient
and Alfven velocity [8] which controls the reacceleration (see Chapter 3) level. (2)
Upon getting a good region in the propagation parameter space, run through the
whole WIMP search analysis with different propagation parameters to quantify the
uncertainty in the dark matter sensitivity resulting from the propagation models.
In the end, PAMELA results and prospects for future experiments on indirect dark
matter searches, especially AMS02, are also briefly discussed.
Chapter 2
Dark Matter
2.1
Evidence for Dark Matter
The first hints of dark matter arose from the study of a cluster of galaxies. Galaxy
clusters are the largest gravitationally bound systems in the universe. In 1933, Fritz
Zwicky [67] applied the virial theorem to the Coma cluster of galaxies and found
that the mass estimated in this way was about 400 times larger than that estimated
based on the brightness of the cluster. Even counting the X-Ray gas which makes up
- 50% of the baryonic mass of the cluster, there is still far less mass than that deduced
from the gravitational effect. A lot of evidence thereafter has been accumulated in
the favor of the existence of dark matter, which infers its existence mostly only via
gravitational force. One galactic scales, the observations of the rotation curves of
galaxies reveals the existence of unseen matter. The roatation curve describes the
orbital velocities of stars and gas as a function of their distance from the galactic
center.
In Newtonian dynamics the circular velocity is v(r) =
G
, which is
expected to be falling oc 1/fr beyond the galactic bulge or visible disks. However
observed rotation curves [60] exhibit a characteristic flat behavior at large distances,
even far beyond the visible disks. For example, in our own Galaxy, v
220 km/s
at the location of our solar system (8.5 kpc) and remains largely constant out to the
largest observable radius (- 50 kpc). Flat rotation curves can be explained by the
presence of a dark halo with density p(r) oc 1/r 2 , so that M(r) oc r.
Recent N-body simulations suggest the existence of a universal dark matter profile
[52]. A generic parametrization of a dark matter halo density is
Po
(r/R)(1 + (r/R)) (0-7
)/ a
Some widely used halo profile models include the isothermal (Iso) [19], Navarro, Frenk
and White [52], and Moore, et al [49]. Their values of the parameters (R, a, , -y)
are given in Table 2.1. I use isothermal profile in this analysis, which is the most
conservative one in terms of annihilation rate. The most recent estimate of our local
dark matter density, po, assuming an isothermal profile, is Po - 0.3 GeV/cm3 [38].
There is, however, significant uncertainty in po. For example, this value may be
Model
Iso
NFW
Moore
a
2.0
1.0
1.5
3
2.0
3.0
3.0
7y R(kpc)
3.5
0
20.0
1.0
28.0
1.5
Table 2.1: Parameters of some widely used profile models for the dark matter density
in galaxies.R is the galactic core radius, and can vary from galaxy to galaxy
increased up to a factor of 2 due to the detailed model of our Galaxy [38]. I use
Po = 0.3 GeV/cm3 in my analysis.
The joint analysis of the anisotropy of the cosmic microwave background (CMB),
large scale structure and Type Ia supernova data [44], provide stringent constraints
on the constituents of the mass-energy budget of the Universe: baryons, dark matter
and dark energy. The results indicate that neutrinos are far from enough to account
for all the dark matter [56], and that dark matter is cold or warm, and non-baryonic
[21]. The dark matter is now known to be about 22% of the total energy density in
the universe (QDM " 0.22)[42], which provides an important constraint on theoretical
modeling of dark matter candidates.
2.2
Dark Matter Candidates
Candidates for DM must be non-baryonic, cold or warm, and stable over cosmological
time scales, which leave open many possibilities. There are numerous plausible DM
candidates that have been discussed in the literature ([21], [26] and references therein),
whose masses and interaction cross sections span many orders of magnitudes, as shown
in Figure 2-1.
2.2.1
WIMPs
Weakly interacting massive particles (WIMPs) are among the most well-motivated
DM candidates. WIMPs are particles with mass mix roughly between 10 GeV to
several TeV, and with cross sections near the weak scale. Their present relic density
can be calculated reliably assuming WIMPs are in the thermal and chemical equilibrium with the hot soup of Standard Model particles after inflation. When their
interaction rate drops below the expansion rate of the Universe, WIMPs drop out
of thermal equilibrium ("freeze out"), and their co-moving density remains constant
afterwords. Freeze-out happens at temperature TF r mx/ 2 0, which implies that
WIMPs are already non-relativistic (cold) when decoupling from the thermal plasma.
An order-of-magnitude estimate of the relic density of WIMPs is given by [37]:
O.lpb. c
xh2 ~0
< 0-AV >
For < oAv >= 1 pb.c, typical for the weak scale, this thermal relic density has the
right order of magnitude required by cosmology (QDM _ 0.22). A leading WIMP
candidate is the neutralino, a SUSY particle.
Some Dark Matter Candidate Particles
i0
.040
10
neutrinos
photon
oKK
'if
Ws
zyDM
t sperWIMPs:
g
no
particle
Model of
by the
unsolved
remaining
u that
Alot of problems
of a
limit
the low-energy
maycurrent
be onlyStandard
Model
indicate
our Standard
physics
cross-section
on a mass versus inter action
dard Model. Mandy featureses
extensions of the Stanus
32
is its role in understanding the hierarchy problem - why the weak force is 10
times stronger than gravity. The question is actually why the Higgs boson is so
much lighter than the Plank mass. The problem arises in the radiative corrections
to the mass of the Higgs boson. At 1-loop, the correction scales as Sm,2
(a )A,
important
play
an
to
become
physics
where A is a high-energy cut-off where new
role. If we take this scale to be the Planck scale, then we have a quadratically
diverging Lagrangian. Supersymmetry provides a solution to this divergence problem.
It assumes that there exists a superpartner of each Standard Model particle, with
the same mass but with spin different by 1/2 (turning bosons to fermions, and vice
versa). Since the contribution of fermion loops to 6m 2 have opposite sign to the
corresponding bosonic loops, the quadratic divergence terms are canceled out, and
the radiative correction at i-loop level becomes [21]
m 2
2w
)(A2
+ mB)
2w
)(A2 +
==F2)
(
2w
)(mB2
-
F2)
Provided mB 2 - mF21 _ 1 TeV, supersymmetric algebra ensures that perturbation
theory is valid up to the Planck scale.
Another reason for interest in SUSY is that it naturally makes the gauge couplings
of strong, weak and electromagnetic forces converge at a scale Mu 12 x 1016 GeV
[10]. Without the contribution of SUSY partners, the extrapolation of the coupling
constants using only Standard Model particles fail to converge to a common value.
Last but not least, Supersymmetry provides an excellent dark matter candidate:
the lightest stable particle, called the neutralino. The superpartners of gauge bosons
B,W3 (or photon and Z, equivalently), and the neutral Higgs bosons H 10 , H20 , mix
into four Majorana fermionic mass eigenstates.
-0
~--0
x = N 11 B +
N
12 W 3
- N
2 1 HI
+ N
22
H2
The neutralino is the lightest mass eigenstate. Provided R-parity [65] is conserved,
the neutralino is stable and can only be destroyed via pair annihilation (because
the neutralino is a Majorana particle, which means it is its own anti-particle). As
discussed before, as long as their self-annihilation cross section is of weak strength,
neutralino relic density can account for the cosmological DM. The leading channels
for neutralino annihilations are annihilations to fermion-antifermion pairs (primarily
heavy fermions, such as bottom and charm quarks and tau leptons), gauge bosons
pairs (W+W-, ZZ) and final states containing Higgs bosons. For a complete list of
all tree level processes, see [37].
2.3
(c)MSSM and Benchmark Points
The Minimal Supersymmetric Standard Model (MSSM) is the minimal supersymmetric extension of the Standard Model, in the sense that it contains the smallest
possible field content necessary to give rise to all the fields of the Standard Model. We
associate fermionic superpartners to all gauge fields (Gluons,W±, Z, y), and scalar superpartners to all fermions (quarks and leptons). Finally, two Higgs superfields need
to be introduced, up-type Higgs doublets and down-type Higgs doublets, where only
one is required in the Standard Model. An extra one is necessary because it is impossible to introduce Yukawa couplings of both up and down type quarks to the same
Higgs field in the MSSM [37]. Furthermore, introducing the second Higgs field can
make the theory anomaly-free [53]. Table 2.2 is a summary of the resulting particle
Superfield
SM particles
Spin
Q
d
1/2
Uc
un
De
Superpartners
Spin
1/2
un*
0
dR
1/2
dR
0
L
L
eL
1/2
eL
0
Ec
HI
H2
G"
en
H1
H2
g
1/2
0
0
1
Wi
B
Wi
B
1
1
eR*
H1
H2
g
W
B
0
1/2
1/2
1/2
1/2
1/2
Table 2.2: Field content of MSSM
content of MSSM.
The MSSM has 105 free parameters. Practical phenomenological study of the
MSSM requires that the number of parameters must be reduced by making some
theoretically well-motivated assumptions. The minimum supergravity (mSUGRA)
scenario [39], a constrained MSSM (cMSSM), is a popular phenomenological model,
in which the gauge couplings, the soft supersymmetry-breaking scalar (sfermion and
Higgs boson) masses mo, gaugino masses ml/ 2 and trilinear couplings Ao are each
assumed to be universal at the Grand Unification Scale. The 105 free parameters are
thus reduced to five (four continuous and one sign) in this scenario:
mo, ml/ 2 , A 0 , tan(3), sign(p)
where tan() is the ratio of the vacuum expectation values of the two Higgs fields
and p is the higgsino mass parameter.
These parameters are constrained by many experiments, including supersymmetric particle searches at collider experiments [47], rare B decays [36], the measurement
of the anomalous magnetic moment of the muon [18], and the relic dark matter Q2DM
measurement from WMAP and other cosmological data taking [42] [44]. A set of
benchmark scenarios in the framework of the mSUGRA , taking into account all the
constraints mentioned above, was proposed by Battaglia et.al [17]. These benchmarks
(see Table 2.3) allow clear comparison of many different experimental results. In my
analysis I follow them to survey the dark matter parameter space.
2.4
DM Search Methods
Many methods have been proposed to search for WIMPs. These include experiments
which hope to measure the recoil of dark matter particles elastically scattering off of
nuclei (direct searches) [28], experiments such as AMS01, which hope to observe the
Model
m 1 /2
mo
tan/3
A
600
120
5
B
250
60
10
C
400
85
10
D
525
110
10
E
300
1530
10
F
1000
3450
10
G
375
115
20
H
935
245
20
I
350
175
35
J
750
285
35
K
1300
1000
35
L
450
300
50
M
1840
1100
50
sign(p)
+
+
+
-
+
+
+
+
+
+
-
+
+
mt
X mass
175
252
175
98
175
163
175
220
171
115
171
430
175
153
175
402
175
143
175
320
175
573
175
187
175
821
Table 2.3:
Post-WMAP cMSSM benchmark parameters sets proposed in [17].
mo, ml/ 2 , mt and X are in units of GeV. Ao is assumed to be zero for all scenarios.
products of dark matter annihilations (indirect searches), and collider experiments
[15] which hope to produce DM particles. Indirect searches include searches for neutrinos [34], gamma-rays [50], and charged final states [16] [27]. The fluxes of these
annihilation final states are proportional to the annihilation rate, which in turn depends on the square of the dark matter density. Neutralino annihilation produces
the same amount of electrons and positrons, protons and antiprotons respectively.
Positrons and antiprotons, however, have low background from known astrophysical
sources. Therefore, they are better channels than electrons and protons for dark
matter search. However, in practice, it is highly challenging to measure the positron
or antiproton energy spectrum accurately. The challenge mainly comes from distinguishing positrons from protons, knowing that the abundance of protons in cosmic
rays is about 104 times of that of positrons, so even a 10 - 4 leakage of protons in
positrons would mess up the positron spectrum completely. The same reasoning also
applies to antiprotons. AMSO1 positron ID was limited to a momentum range of up
to about 3 GeV due to the Cerenkov limit [5]. Thus electron channel is chosen instead
for the analysis. An upper limit on the dark matter annihilation rate can be set by
requiring that the associated electron emission does not exceed the observed flux.
2.4.1
DM Search w/ AMSO1
Electrons and positrons can be produced in a variety of dark matter annihilation
modes. For example, annihilations to ZZ or W+W - can produce electrons and
positrons with energy of mx/ 2 . A continuum of electrons and positrons, extending to
much lower energies, can be produced in the cascades of annihilation products such
as heavy leptons, heavy quarks, Higgs bosons and gauge bosons. The spectrum of
electrons or positrons produced in dark matter annihilations significantly depends on
the annihilation modes, which directly associate to the supersymmetric scenario used
to construct the neutralino.
Electrons and positrons move under the influence of the galactic magnetic fields,
thus they will loose memory of their origin. Diffusion is usually employed to model this
transport process. We need to take into account the annihilations within a spherical
volume centered at the observer (the Earth) and with radius equal to the propagation
length of the electron. The modeling of such propagation in galaxy is discussed in
detail in the next chapter. Figure 2-2 shows the energy loss time-scales of nucleons
electrons & positrons
nucleons
10
/
1 GeV
Kinctic cner
1 TeV
10y
per nucltm. McV
GeV
Kinetic cner.
1 TeV
McV
Figure 2-2: The left panel, time scales of energy loss for nucleons in the interstellar
matter; the right panel: for electrons and positrons.
matter [621. Mainly resulting from the inverse Compton scattering on both starlight
and the cosmic microwave background and Synchrotron radiation, it takes about 100
million years for the energy of an electron to drop from 100 GeV to 1 GeV. 1 GeV is
approximately the minimum energy of our detection range. The diffusion coefficient D
is typically 1028 cm 2 sec-1 [62], thus the propagation length of the electron is DT
3
kpc. The lifetime of antiprotons is about 5 orders of magnitude greater than that of
electrons, resulting 2 to 3 orders of magnitude greater in their propagation length.
Therefore, electrons detected by AMS01 have their origins within 3kpc of Earth, while
anitprotons originate from the whole Galactic halo.
2.5
Substructure of DM Distribution
There are reasons to believe that the dark matter distribution in the Galactic halo is
not smooth. The existene
existence of "clumps" is a common feature of N-body simulations.
There are theor
theoretical arguments to support this result. For instance, initial density
fluctuations after the Big Bang are expected to produce clumps of dark matter at all
scales [22]. If cosmic strings or other defects exist, they may seed the formation of
density-enhanced dark matter clumps [41]. Also, we know that the Galaxy has consumed smaller dwarf galaxies that have dark matter. The should produce anisotropies
in DM distribution in the Milky Way [1].
The electron flux from neutralino annihilation could be enhanced by the substructures in the galactic halo. If we assume we are averaging over a volume 13 where 1
is the electron propagatonn length, containing many small clumps, which have scales
much smaller than the propagation length of electrons, we can define an enhancement factor, which is called the boost factr. The boost factor, B, is defined as the
ratio of the average of the squared density of dark matter with a clumpy distribution to that with a smooth isothermal distribution, B = < p 2 >clumpy /< 2 >so. It
can satisfy self-consistency to introduce a boost factor for electron and positron flux
because there is actually freedom to have clumpy distribution models to accommodate electron and positron flux while still producing the right Galactic rotation curve
and will not overproduce other annihilation products, such as antiprotons, neutrinos
and gamma rays. The electron flux from annihilation scales as < p2 >, while the
rotation curve depends on < p >. Figure 2-3 is a simple example to illustrate that
< p2 > could be enhanced while < p > remains the same when the distribution
becomes inhomogeneous. Because the propagation of electrons or positrons, antiprotons, neutrinos and gamma rays are not the same, the boost factors for them are not
necessarily the same. Gamma rays and neutrinos are charge neutral and they don't
interact with the galactic magnetic fields, so their fluxes are obtained by integrating
along the line-of-sight connecting the Earth and the source. For antiprotons, they are
similar to the case of electrons and positrons in terms of integrating over a volume.
However, a nearby clump would serve to increase the electron and positron flux more
than it would increase the antiproton flux, as electrons and positrons have shorter
propagation length (roughly several kpc) than that of antiprotons because of their
rapid energy loss [20].
In 1994 and 1995, the High Energy Antimatter Telescope (HEAT) observed a flux
positrons well in excess of the predicted rate from astrophysical sources.
cosmic
of
The energy spectrum peaked at 10 GeV and extended to higher energies [16]. Such
an enhancement could be explained by a clumpy dark matter halo with boost factor
(from several tens to ten thousands, vary as SUSY models) [12].
p= O
Figure 2-3: In the left panel, the density distribution is homogenous over the whole
unit area, thus < p >= 1, < p 2 >= 1; In the right panel, all matter concentrates on
the left half of the unit area, thus < p >= 1, < p2 >= 2, so the boost for this case is
2.
Chapter 3
Cosmic Rays
3.1
Overview of the Galactic Cosmic Rays
The cosmic rays incident at the top of the terrestrial atmosphere include all stable
charged particles and nuclei. There are magnetic fields in the galaxy, which is on
the order of 10-' Gauss. A charged particle moving in the magnetic field executes
a helical path. Using the relation between the momentum P, magnetic field B and
gyroradius rg,
p1 (MeV/c) = 3 x 10-4Brg(Gauss -cm)
a 1014 eV/c proton would have a radius of 3 x 1018 cm, or 1pc. This is much smaller
than the typical scale of a galaxy, which is about 10 kpc. Therefore, cosmic rays below
this energy can be seen to conduct a random walk and are confined by the magnetic
fields in the galaxy. Even though they have no memory of their origins when they
reach the Earth, they are of Galactic origin. This leads to the assumption of a steady
state condition on a galactic scale. The renewal of the cosmic rays requires an average
energy input to compensate for the loss of nuclei by nuclear and electronic collisions
or escape from the galaxy. The lifetime of the confinement in the interstellar magnetic
field is
-
10' yr, as measured by the surviving fraction of the radioactive secondary
nuclei Belo [58]. If we take the galactic radius as 10 kpc, we have a volume of 10"6
cm 3 . The energy density of cosmic rays is about 1 eV/cm 3 . Therefore, the energy
production rate in cosmic rays should be leV/cm3 x 10 68 cm 3
107 yr - 104 2 erg/sec.
The explosion energy of a supernovae is about 1051 erg and the event rate is about
once per 100 years. Thus the energy production rate matches the required renewal
rate of the cosmic rays estimated above. Therefore, it is commonly believed that
supernovae remnants is a principal mechanism for generating and accelerating cosmic
rays, which is also supported by many arguments based on the chemical patterns [58].
"Primary" cosmic rays are those particles accelerated at astrophysical sources and
"secondaries" are those particles produced in the interaction of the primaries with the
interstellar matter. Electrons, protons, helium nuclei, as well as carbon, oxygen, iron
and other nuclei synthesized in stars, are primaries. Nuclei such as lithium, beryllium,
boron, which have low binding energy and thus would be destroyed during stellar
burning process, are secondaries. Figure 3-1 shows the comparison of the chemical
108
107
Solar System
-o-
10
GCR
10 He
o
10
-~
10,
S10
e
20
Fe
Si
Ar
Be
Ti
> S1000
)I 10-1
Zn
10-2
10-3
.........
..............
10
0
5
10
15
20
25
30
Atomic Number (Z)
Figure 3-1: Comparison of the chemical distributions in the Solar System and in the
cosmic rays. The Solar System abundances in the figure represent the abundances of
elements in the proto-solar nebula. [48]
distributions in the stars and in the cosmic rays. The most dramatic difference is a
six orders of magnitude enrichment in the cosmic ray abundances for the elements
LiBeB, which indicates that these light elements were born mainly during the course
of propagation.
The energy distribution of cosmic rays from several GeV to 100 TeV is smooth,
with a power-law distribution [54]. The energy spectra of several components of
cosmic rays are shown in Figure 3-2. Fermi acceleration [30] by the supernovae
shocks can naturally yield a power-law structure. The propagation is a diffusion
process, which will largely preserve the power-law nature, although the index may
change. The modeling of propagation is discussed in more detail in Section 3.2. The
roll over at lower energy results from solar modulation, which is discussed in Section
3.3.
3.2
3.2.1
Propagation of the Cosmic Ray Nuclei in the
Galaxy
Cosmic Ray Propagation Model
Diffusion theory is often employed to describe the large-scale galactic propagation
of the cosmic rays. The cosmic rays move in spiral trajectories around the galactic
magnetic fields with gyroradii determined by the particle energy. The collision against
the irregularities and moving magnetic fields diffuse the cosmic rays [30] [58]. This
provides the mechanism of the confinement and acceleration of the cosmic rays in
0.1104 ,~
10 -
+ He
o
da
2
S
Fe
W
10-74
10-8
10-5'
*4
f4
10-9
.
10
102
. .
103
,
0
.. .
104
.
. .
105
, , .,
106
,
107
Kinetic energy (MeV/nucleon)
Figure 3-2: Cosmic ray nuclei spectra, from PDG Cosmic Rays, T.K. Gaisser and T.
Stanev.[54]
the Galaxy. The moving magnetic fields are known as Alfven waves [8]. The velocity
Alfven waves propagate along the Galactic magnetic field is called Alfven velocity VA,
which can be estimated as VA = B/p 1/ 2 [8], where p is the interstellar matter density.
For more details see Appendix A. p is 1 proton/cm3 , and B is 10- Gauss, thus VA
is 20 km/sec. Cosmic ray particles gain energy from head-on collisions with Alfven
waves and lose energy from overtaking collisions. The head-on collisions happen more
frequently than overtaking collisions due to the greater relative velocity, which results
in an average net gain. Thus scattering by Alfven waves provides a mechanism of
cosmic ray reacceleration.
We consider the propagation of cosmic rays in the interstellar medium with ran-
dom hydromagnetic waves. The transport equation has the following form [62]:
-- (, p, ) = q(F,p)+z.[DxxV
at T=p
-VW]+
a 2
[p2
p
a
(p)
2
p p2
-
a dp 1
p
[(
dt
-
3
V- )
]
TT
fT
(3.1)
2
where IFis the density per unit of total particle momentum, '(p)dp = 4rp f(p) in
terms of phase-space density f(p. q(F,p) is the source term, including the primary
particles injected from supernovae remnants and the secondary particles produced
during the course of propagation. The time interval between supernovae explosions
in galaxy is much smaller than the lifetime of cosmic rays, so supernovae explosion
can be seen as a continuous process, which adds source constantly into cosmic rays.
Therefore, the source term q(F,p) is not a function of time. The radial distribution of
the sources is chosen to reproduce the cosmic-ray distribution deduced from EGRET
> 100 MeV gamma rays [61]. The injection spectrum is assumed to be a power law
in energy resulting from Fermi acceleration. Dxx is the spatial diffusion coefficient.
V is the advection velocity, which is assumed to be in the direction perpendicular to
the galactic plane only and increases linearly with the distance from the plane [66].
Reacceleration is described as diffusion in momentum space and is determined by the
coefficient Dpp. Various losses are also taken into account, including momentum loss,
fragmentation and radioactive decay with time scale tauf and 7, respectively.
We use the software package GALPROP [62], which solves this transport equation numerically given a source distribution and a free escape boundary condition,
to perform the propagation calculation. The model assumes a cylindrical shape of
the galaxy with a radius of about 20 kpc and the total height of several kpc. The
primary sources are distributed in a thin disk about the galactic mid-plane having
characteristic thickness of 200 pc.
Modeling galactic propagation of cosmic rays is a very complicated task. The
major sources of the uncertainties are fourfold: (1) primary source distribution and
the interstellar matter distribution; (2) cross sections for secondary cosmic ray production, annihilation and scattering; (3) galactic propagation models and parameters;
(4) solar modulations. The uncertainty we try to investigate is from (3), while for
the other three, we use the values to the best of our knowledge. See Ref. [62] [51] for
more details for (1) and (2). The solar modulation is discussed later in this chapter.
3.2.2
B/C -
A Probe of the Propagation Model
Secondary nuclei are produced by the interaction of primary nuclei with the interstellar medium during propagation. Therefore, their spectra carry the information of the
propagation process. Taking the ratio of the secondaries to the primaries can largely
cancel out the dependence on the injection spectra, leaving the quantity very sensitive
on the variation of the propagation parameters. B/C is the best secondary-to-primary
ratio for this purpose, not only because they are relatively well measured in cosmic
rays, but also because the cross sections for the production of their secondaries are
well known.
More energetic particles tend to escape from the galaxy faster. However, on the
other hand, particles of higher energy tend to be scattered with the magnetic fields
more often, thus would spend more time in the system. The competition of these two
mechanisms results in the shape of the B/C spectrum, which rises with energy at low
energy but decreases at high energy. Therefore, the B/C ratio peaks at the energy
where the dominant mechanism transitions from one to another.
Diffusion, reacceleration and advection are the three propagation mechanisms in
the transport equation. Diffusion determines the overall height of the B/C spectrum,
since the sooner the primaries escape from the galaxy, the fewer secondaries would
be produced (see Figure 3-3).
The reacceleration, introduced by the moving magnetic fields, results in more
frequent collisions. This changes the energy at which escape balances collisions. Thus,
the Alfven velocity determines the peak position of the B/C spectrum (see Figure 3-4).
Measurements indicated that the B/C spectrum peaks at 1 GeV/nucleon [62].
An outflow of material from the halo could carry the cosmic rays with it. This
advection motion would affect the B/C spectrum. Advection is therefore adds a third
mechanism that affects the escape time. The outward advection would speed up the
escape of cosmic rays from the Galaxy. Thus advection would reduce the overall
height of the B/C spectrum and shift its peak towards lower energy (see Figure 3-5).
B/C with different diffusion
o0.45
0.4
0.35 0.3
.----- ..
---
"
S.
Dxx=5.00e28 cm2s -1
-
Dxx=5.75e28 cm 2 sl
Dxx=6.50e28 cm2s -1
0.250.20.15 0.1
-'"...."
0.05 -..
.1
.
"'-
10
100
1
Momentum (GeV/c) per nucleon
1000
Figure 3-3: The effect of diffusion on B/C, as simulated by GALPROP
Although in principle, all three processes could be important, we only consider diffusion and reacceleration (DR model), since it is the minimum combination which can
reproduce the key observation of B/C. The effect of advection on the B/C spectrum
B/C with different reacceleration
o0
m-
.45
0.4
V....
A=20 km s -
-
0.35 0.3 .o.°
-o, .
...1
..
-
VA= 3 6 km s 1
...--
VA=50
km s'
A-
m.
0.25
-
I
rtmt
I
we
mr
0.2
rm
0.15
atrr
rr, m
mmmmr*
=um
i
nr'mlImm:
0.1
0.05 F
I
1
I I 111 I
I
I I I 1111
I
I
111111I
m
m
m Lm..U
1
10
100
Momentum (GeV/c) per nucleon
1000
Figure 3-4: The effect of reacceleration on B/C, as simulated by GALPROP
B/C with different advection
o0 .4 5 -0.35
---- dvdz=l0 km s- kpc -
0.35 0.3
0.25
0.2
dvdz=0 km s-1 kpc
-
O0.4
.. dvdz=20 km s- kpc1-
II
Im
Ir Ir r~
r Ir
tUrl.tltWlrf
itt4ltl~t(yl
*' ~i
iur1~
Irt~
"'
,,
Ir
,,rt
0.15
I
I
Ir
rr
rrrrrcrr ri
I
rnlr II ,
rir,rrll Ir
rrll rr rr I
Ir~r
I
rrii II
I
Ir~r"
ritt~ri
0.1-
rrrriii
ri iiri
"r,,,,
l)llnll
"'"~i~b~~
0.05 D
0.1
,,,,,I
', ', '""'
,
1
,' ', '""'
, ,,, I
10
.
, I ,,,I
', I' '""'
100
RAk~ka
' ' '""
1000
Momentum (GeV/c) per nucleon
Figure 3-5: The effect of advection on B/C, as simulated by GALPROP.
24
can be produced by varying diffusion and reacceleration.
3.2.3
Propagation Parameters
The key parameters for diffusion and reacceleration are D,, and Dpp, respectively.
They can be derived from scattering with Alfven waves (see Equation (9) (10) in [57]).
Dxx can be formulated as a power law in particle rigidity
DXX = DxxoR'
where 3 = v/c, R = p/Z. For Alfven waves with a Kolmogorov spectrum, 6 = 1/3
[57]. Dp can be described by a simple approximation [62]:
Dpp = p2VA/(9Dxx)
where VA is Alfven velocity. Therefore, Dxxo and VA are the key parameters in this
framework. Due to the poor knowledge of the interstellar medium and the lack of
direct measurement for these parameters, their values must be obtained from fitting
the cosmic ray data. One major accomplishment of this work is to constrain Dz0o
and VA from AMSO1 measurements of the B/C spectrum.
3.3
Solar Modulation and Geomagnetic Effects
The cosmic rays are modulated by the solar wind, the expanding magnetized plasma
generated by the Sun, which decelerates and partially excludes the lower energy particles from the inner solar system. We make use of the forced field approximation
suggested by Gleeson and Axford (1968) [31].
E2 - E2
J(r,E) = (E +
)2-EJ(r,
E+
(3.2)
where J(r,E) is the differential flux of cosmic ray, r is the distance from the Earth,
E is the total energy of the particle, Eo0 is the rest energy of the particle. This model
is characterized by one parameter D, which can be interpreted as the energy loss of
the cosmic-ray particle when approaching the Earth from infinity. 4 = IZleo, where
¢ is the potential for unit charged particles. During AMS01 flight, ¢
0
0.58GV [55].
Cosmic rays are also affected by the Earth's magnetic field. Low energy particles
from outer space cannot penetrate through except at high latitude. Therefore, the
events observed by AMSO1 with energy lower than some limit would most likely be
those trapped by the Earth magnetic field, not from cosmic origin. With a displaced
dipole assumption of the Earths magnetic field, the lowest momentum accessible to
our detector in low Earth orbit is given by [24]:
p
|Z I
_
4
59.6[GeV/c 2] cos A
1 2 2
3
(1 + (1 - Q cos 0 cos ) / )
where A is the latitude of the detector, 0 is the east-west component of the zenith
angle of the incident trajectory. We take care of this geomagnetic cutoff effect in the
exposure time.
Chapter 4
AMS Experiment
The AMSO1 experiment was a magnetic spectrometer designed to measure cosmic
rays from several hundred MeV to about 300 GeV, which was a prototype of the
future experiment AMS02. It flew on Space Shuttle Discovery mission STS-91 in
1998 for 10 days and recorded about 100 million events. In this chapter, section 4.1 is
briefly describe each component of the detector. A quick review of the flight is given
in section 4.2. For detail information of AMS01, please refer to [6].
4.1
The AMSO1 Detector
AMSO1 detector mainly consisted of a permanent magnet, a Time of Flight system
(TOF), a silicon tracker, an Aerogel Threshold Cerenkov Counter (ATC), and an
Anti-Coincidence Counter (ACC). A schematic of the detector is shown in Figure
4-1. I mainly use the information from the TOF and the Tracker for my analysis.
4.1.1
The Magnet
The design of the magnet should maximize the bending power while keeping the magnet dipole and flux leakage small so not to add unwanted torques to the space shuttle
or to affect the electronics. A cylindrical permanent magnet design, constructed of
high grade magnetic Nd-Fe-B blocks, was chosen, which had a length of 800mm, and
an acceptance of 0.82 m 2 sr [6]. With a clever arrangement of the magnetic building
blocks, most of the magnetic flux was returned via the magnetic blocks, reducing
the flux leakage and the dipole moment to acceptable levels. The magnet provided a
bending power of 0.15 T m2
4.1.2
TOF
The time of flight (TOF) had four layers. Two layers formed a double-plane on
top of the magnet, orientated orthogonal to each other to obtain a two-dimensional
measurement. The other two formed a double-plane in the same way below the
magnet. A double plane assembly is shown in Figure 4-2. Each layer consisted of
Particle Trajectory
Low Energy Particle Shields
TOF Lae
T
82
- Tracker T1
Planes T2
/
-
T3
I-~----_"
T4
--
TLi-LI Q
i I I1
-1- - - -*
Cernkov
Figure 4-1: The schematic of AMS01 detector
14 scintillator paddles of 10mm thickness, 110mm width and various lengths ranging
from 720 to 1360mm. Three photomultiplier tubes were attached at each end of the
paddle.
TOF determined two quantities, which were the traversing time of a particle
through the detector, and the direction of the particle, whether it was going upwards
or downwards. It also provided the fast trigger and a rough charge measurement and
trajectory reconstruction.
The time resolution of TOF was better than 120ps which resulted in a position
resolution of better than 2 cm along the paddle, and a velocity resolution of - 3%.
A highly relativistic (1 = 1) particle would take 5 ns to traverse the TOF system,
therefore the upward and downward going particle separation was expected to be 1
in 1011 [11]. More information of TOF can be found in Ref.[11] [9].
4.1.3
Tracker
The tracker had six layers. Each layer consisted of ladders made of seven to fifteen
double-sided silicon microstrip sensors with dimension 40.14 x 72.04 x 0.30 mm3 .
Figure 4-3 is an exploded view of the ladder. The sensors were read out from both
sides, which ran perpendicular to each other. The S-side measured position in the
bending plane with a pitch of 110 pm, and the K-side measured position in the
non-bending plane with a pitch of 208 pm. The ladders were aligned parallel to
5 pco4Foo
Figure 4-2: The schematic of a double-plane assembly of AMS01 TOF.
the magnetic field to maximize the position resolution in the bending plane. The
materials in each plane corresponded to approximately 0.65% of a radiation length
and - 0.06% of a nuclear interaction length.
The tracker provided a measurement of the rigidity and charge of a particle. It had
a position resolution of about 10 micron in the bending plane and about 30 micron in
the non-bending plane, which could be translated to a momentum resolution of 9%
for protons of 10 GeV. It had a theoretical rigidity range of about 100 MV to 600
GV. However, multiple scattering (MS) in the silicon tracker material would introduce
deviation in the particle tracks which should be considered in the error estimation.
The mean change in the trajectory due to MS in the Gaussian approximation is given
by [33]:
eV zx/Xo(1 + 0.0381n(x/Xo))
3cp
where p, /c, and Z are the momentum, velocity and charge, and x/Xo is the thickness
of the scattering medium in radiation lengths. It is important to note that 00 oc 1/0,
which means lower energy particles suffer bigger uncertainty from MS. Therefore in
practice, our accessible energy region would be reduced by the multiple scatterings
at low energy and low statistics at high energy.
S13.6=
A measurement of total energy loss in the tracker was used to determine the
absolute charge of the particle passing through the tracker. A combined charge
measurement of the TOF and tracker was estimated to be accurate to 1 in 107 in
distinguishing Z=1 and Z=2 nuclei. Only tracker was used for charge measurement
for higher charged nuclei (Z > 2), because the dynamic range of the TOF scintillator
does not permit to distinguish such large charges with great accuracy. The tracker
could provide good charge measurement for nuclei up to Oxygen. More information
of tracker can be found in Ref. [7].
SEPRSM
Figure 4-3: An exploded view of AMSO1 tracker ladder.
4.1.4
The Aerogel Threshold Cerenkov Counter (ATC)
An ATC was installed underneath the lower TOF layer, which was used to increase
the discrimination power between protons and positions below 3.5 GeV. Since it was
not used in this analysis, for a detailed discussion on its design, construction and
performance, please refer to Ref. [13] [14].
4.1.5
The Anti-Coincidence Counter (ACC)
16 anticoincidence scintillator counter paddles were arranged like barrel staves in the
space between the inside wall of the magnet bore and the support shell of the tracker,
which provided a tag for particles coming from aside. An event would be discarded
if ACC fired since they were unlikely able to be well reconstructed.
4.2
The AMSO1 Flight
4.2.1
Flight Parameters
The flight of AMSO1 on board the Space Shuttle Discovery was from June 2 to June 12,
1998. An important parameter of the flight was called zenith angle, which was defined
as shown in Figure 4-4. The zenith angle was the angle between the pointing direction
of the detector and the line connecting the center of the Earth to the shuttle/AMSO1.
Therefore, a zenith angle of 0Omeans that AMSO1 was pointing up right; a zenith
S.angle
Figure 4-4: The definition of zenith angle.
angle of 1800 means that it was pointing to the Earth.
As shown in 4-5, the data taking could be divided into 4 periods: 1. The 25 hours
before docking with Mir with the zenith angle within 450 . 2. The four days during
which Discovery was docked with Mir. 3. After the Mir undocking, 19, 25 and 20
hours with zenith angle of 00, 200 and 450, respectively. 4. Approximately 9 hours
with zenith angle of 180' prior the descent of Discovery.
The zenith angle fluctuated quickly during period 2, which could be because that
many particles might have interacted with the MIR before passing through the detector. We did not use the data from this period, but one can manage to reconstruct
the layout of MIR seen by AMSO1 from these data [35]. In period 4, AMS was facing
towards the Earth. Because we are interested in those of cosmic origin, we do not use
the data from this period either. Therefore only data taken from the first and the
third periods are used in the analysis, which leads to roughly 94 hours of active data
taking.
4.2.2
Trigger and Data Acquisition
There were three levels of the electronic trigger of AMS01: Fast, Level-1 and Level3 [35]. The Fast trigger required coincident signals from the top and bottom TOF
planes within 200ps. Level 1 triggers required that ACC did not fire, and only specific
combinations of the two outer TOF planes were acceptable to avoid particles hitting
the unequipped parts of the tracker. Level 3 trigger required that at least 3 clusters
-180
m__
160
4
2
'140
4 120
4.z
-
c 100
:-
:
...IT
:',i~,
:J
i
-
N
80
I,'
60
....
..
-
t'.'A
.p,,
.'
i.
3
m~a~
40
20
nI
I
"i l
I
I
I
100
I
I
150
I
200
250
Flight Time (hours)
Figure 4-5: Zenith angle vs Flight time for AMS01 during Space Shuttle Discovery
mission STS-91 in 1998.[24]
on 3 different tracker layers with signal-to-noise ratio above a certain level were along
the expected particle trajectory.
During the flight, events passing all three levels of triggers were recorded. A set of
events that triggered only the fast trigger were recorded for trigger efficiency studies.
These events constituted about 0.1% of the total dataset and were known as prescaled
events. The trigger rate varied between 100 Hz and 1600 Hz depending on magnetic
latitude. The detector was saturated when the shuttle was in the South Atlantic
Anomaly (SAA) where the trigger rate spiked to almost 20k Hz [5]. This extremely
high particle flux is because of the extremely weak geomagnetic fields of this region.
The detector lifetime has to be considered during the efficiency calculation by dividing
the count rate with the corresponding lifetime. Since the lifetime during SAA was
zero, data taken during this period were discarded.
Chapter 5
WIMP annihilation signal search
procedure
This chapter outlines the procedure to prepare major ingredients needed for the search
of WIMP annihilation in AMSO1 Z=-1 data, based on Ref. [55]. Section 5.1 reviews
how to produce the momentum spectrum of negative unit charge particle from detector data. Signal production from neutralino annihilation is described in Section
5.2.
5.1
5.1.1
AMSO1 Z=-1 Momentum Spectrum
Data Selection
Charged particles deposit energy in the TOF and tracker layers when traversing the
detector. This allows for the reconstruction of the particle tracks. Because charged
particles are bent by the magnetic field, the radius of the track allows for the determination of the momentum per charge of the particle, which is defined as rigidity.
Preselection cuts are applied to clean up the data set. There are three categories
of such cuts which are summarized as the following:
1. To make sure events are of cosmic origin, events are discarded if:
* they pass from the bottom to the top or through the side (ACC fired);
* their zenith angle is larger than 50 degrees;
* their energy is smaller than the geomagnetic cutoff (see Equation 3.2).
2. To remove poorly measured data, events are discarded if they have:
* no reconstructed particle track;
* fewer than 4 tracker hits or fewer than 3 TOF hits;
* different signs between the rigidities as reconstructed from the upper 3 layers
and the lower 3 layers of the tracker;
*
different charges as measured by the tracker and the TOF.
3. To avoid detector errors, events were discarded if:
* taken during the time when the detector was at the South Atlantic Anomaly;
* detector lifetime less than 35%.
Because the trajectory through the detector for very high energy particles is almost
a straight line, and the content of protons in the cosmic rays is about 1000 times that
of electrons, a major background for electron events of high energy comes from misidentified protons. Therefore, other cuts are applied to preserve as many electrons
as possible while removing the background protons. Two cuts, CircularCorrection
and Half rigidity difference [55], could serve for this purpose efficiently. CircularCorrecttion requires the rigidities reconstructed from a simple fit of a circular track and
from the FastFit algorithm [7] which accounts for magnetic field inhomogeneities and
multiple scattering not to be too different. Half rigidity difference cut requires the
rigidities constructed using the upper 3 and lower 3 layers of the tracker respectively
not too different. The cutting criteria were determined by maximizing the electron
to mis-identified proton ratio generated by Monte Carlo.
After the cuts, 6,563,097 events are identified as Z=1 particles, and 74,216 events
are identified as Z=-1 particles.
Exposure Time vs. Momentum
1
I
r
1
II
1
I
I
I
I
IIII
I
I
10
Momentum (GeV/c)
I
1 1111
I
I
100
Figure 5-1: Exposure time for protons and electrons.
1000
5.1.2
Exposure Time
The number of counts in each momentum bin must be turned into count rate by dividing by the exposure time. As mentioned in Section 3.3, the Earth's magnetic field
sets a momentum cutoff which varies as a function of the latitude, particle incoming
direction and charge. Therefore, a particle directly from outer space with certain
momentum can only be seen by AMSO1 in a certain region during the flight where
its momentum is larger than the geomagnetic cutoff. The summation of time during
which AMS01 took data within this region is the exposure time for this momentum
bin. Particles with higher momentum have larger exposure time because their momentum is higher than the geomagnetic cutoff over a wider range of latitudes. Because
the geomagnetic cutoff depends only on momentum and the magnitude of particle
charge, the exposure time as a function of momentum for protons and electrons is the
same, as shown in Figure 5-1.
5.1.3
Acceptance
Monte Carlo techniques [63] are used to simulate the gathering power and resolution
of AMS01, producing the acceptance matrices, which translate the cosmic ray flux
(cm-2sr-lsec-1) to the count rate (sec - 1) in the detector. The Monte Carlo events
are generated over a rectangle (250 cmx90 cm), 100 cm above the detector and
are run through the analysis chain as real data. The matrix element (Prec, Pgen)
is the probability of a particle coming from space with momentum Pgen detected
at momentum Prec. The matrix elements have unit of cm 2 sr, which come from
multiplying by the event generating phase space, which is the rectangle area. The
acceptance matrices for protons and electrons are shown in Figure 5-2 [55]. Panel
(a) is the matrix of Monte Carlo protons detected with correct sign charge. It is also
used as an approximation of the acceptance of antiprotons recorded as antiprotons.
Panel (b) is the matrix of Monte Carlo protons recorded as electrons, which is used
to calculate the misidentified proton rate from proton background.
Both systematic and statistic uncertainties are considered in the acceptance matrix. By comparing with the pre-scaled events, it is found that the simulation is over
efficient by 13+3.5 % [5]. Therefore, the count rate in Monte Carlo is reduced by 13%
and a systematic error of 3.5% is set. The statistic error assumes poisson statistics.
If the counts is N, the uncertainty is of v-N.
5.1.4
Background Fit
As shown in the Monte Carlo simulation, high momentum protons are often misidentified as high momentum electrons, and vice versa. Because protons outnumber electrons by a factor of 1,000, misidentified electrons only add a very small fraction to the
proton spectrum while misidentified protons are a significant background of electrons.
GALPROP predicts the cosmic ray backgrounds given the particle injection spectrum and the propagation model. The injection spectra of proton and electron are
assumed to have a power law in energy NE - 7. The normalization and the index are
Protons indentified as protons
Protons indentified as electrons
a)
b)
10lo
E
E 100
a
10
o
0
10
10-
eo
0
-
o
10
11
1
10
100
recorded momentum (GeV)
1
Electrons indentified as Electrons
10
100
recorded momentum (GeV)
Electrons indentified as Protons
1000
1000
c)
d)
E1
E 100
E
00
E
00
o
E
10
10
o
E
10
10
10
1
1
10
100
recorded momentum (GeV)
1
10
100
recorded momentum (GeV)
Figure 5-2: AMSO1 acceptance matrices for electrons and protons [55]. Particles detected with the correct sign charge are also mostly detected with the correct momentum, as shown in panel (a) and (c). High momentum protons are often misidentified
as high momentum electrons, as shown in panel (b).
Proton Background Fit to AMS-01 Data
10
0
0
1
-2
10-
10
Momentum (GeV)
100
Figure 5-3: Proton background fit to AMS data [55].
left as free parameters in the fit. The diffusive reacceleration propagation model with
default parameters setting (see Appendix C) is employed.
All Z=1 events are assumed to be background cosmic ray protons. The spectrum
generated by GALPROP, modulated by solar modulation and multiplied by protonto-proton acceptance (Figure 5-2 panel (a)), is fit to our Z=1 data. The fit yields a
normalization of 0.67 ± 0.02, as a fraction of the initial GALPROP normalization,
and an injection index of 2.43 ± 0.01 with X 2 /N = 7.98 (N = 13) [55] (Figure 5-3).
We then perform the background fit to Z=-1 data, assuming no dark matter
signal. There are two main components. One is the misidentified protons, which
can be calculated using the resultant proton background and the proton-to-electron
acceptance matrix (Figure 5-2 panel (b)). It is kept fixed in the fit. The other
is cosmic ray background electrons, which is treated in the same way as that for
protons. The fit yields a normalization of 0.70 ± 0.04 and an injection index of
2.70 ± 0.04 with X2 /N = 2.04 (N = 13) [55] (Figure 5-4). In principle, Z=-1 data
should include antiprotons. However, it should be safe to ignore this in the fit because
the abundance of antiprotons in cosmic rays is tiny compared to that of electrons,
only about one thousandth.
Components of Electron Background
10-10 10-2
*0 Charge -1 Data
MislDed Protons
Electron Background
Sum of Backgrounds
H
-
10 -3
0
lo1
(U
I
I
I
I
i
I
I
10
I
lIIi
100
Momentum (GeV)
Figure 5-4: Electron background fit to AMS01 data [55].
Electron Background Fit Residuals
10
Momentum (GeV)
100
Proton Background Fit Residuals
10
Momentum (GeV)
100
Figure 5-5: Residuals for AMS01 electron (left) and proton (right) background fit.
[55].
5.2
Neutralino Annihilation Electron and Antiproton Signal Production
As mentioned in Chapter 2, the exact process of neutralino annihilation is model
dependent. We use the benchmark points in cMSSM framework listed in Chapter
2 for this analysis. The electron or antiproton spectrum at Earth from neutralino
annihilation is given by:
ddE
M0
aiuivBi f drdE'f2 (E')G(E',E, r)p2(r)
(5.1)
For a particular cMSSM scenario, DarkSUSY [32] is used to calculate the neutralino
mass m, their self annihilation branching ratios Bi and the cross section ai for each
channel. Then Phythia [59] is used to simulate the spectrum of electrons or antiprotons f (E') produced in the cascades of particles produced in each annihilation
channel. An example of benchmark I is shown in Figure 5-6. An isothermal distribution is assumed for the dark matter distribution pop(r), where the local density Po
takes the value of 0.3 GeV/cm3 . The Green's function G(E', E, r) describes the evolution of a monoenergetic electron or antiproton signal during the propagation from
the place of neutralino annihilation to the Earth, which is simulated by GALPROP.
Figure 5-7 shows an example of the Green's function for a 10 GeV monoenergetic
electron source with isothermal distribution, integrated over the volume with the
scale of electron propagation length: f drG(E',E, r)p(r). Finally, the signal specV
trum at the Earth is obtained by convolving the Green's functions with the original
electron or antiproton spectrum. Figure 5-8 shows the electron and antiproton spectra from neutralino annihilation for SUSY benchmark I before and after the galactic
propagation.
XX -> Electron Spectrum for Model I
E
0l 1
~WW
ESum -
ElO
0
1
1
4
10annihilation
for benchmark
annihilation.
XX -> Antiproton Spectrum for Model I
SSum
10"
[55].
W+W
1
The legend gives the final states of neutralino0
greens function for 10 GeV electrons
S200
'180
U)
160
140
0
1
10
1(
Kinetic Energy (GeV)
Figure 5-7: Green's function for 10 GeV monoenergetic electron source with isothermal distribution.
Spectrum at Earth for SUSY Model I
Energy (GeV)
10
Energy (GeV)
1000
Figure 5-8: Electron and antiproton spectra from neutralino annihilation at Earth
for benchmark I [55].
Chapter 6
The B/C Ratio
This chapter describes the procedure to select boron and carbon events from AMSO1
data and to obtain their acceptance by using Monte Carlo. There are 21,674 carbons
and 8,582 borons selected out of 100 million raw data. A fit on the B/C momentum
spectrum is conducted to constrain the galactic propagation parameters. The diffusion reacceleration model is adopted. The fitting result is given on the propagation
parameter space.
6.1
Charge Identification
The identity of boron and carbon is straight forward. They are distinguished from
other elements by their charge. Boron has charge equal to 5 and carbon has charge
equal to 6. The tracker of AMSO1 has very good charge identification ability up to
oxygen (Z=8). For lower charged particles (Z=1, 2), both TOF and tracker are used
to determine the charge. However, due to the limited dynamic range of the TOF [25],
only tracker information is used in the selection of boron and carbon.
6.1.1
Energy Loss Of Heavy Nuclei
Charged particles lose energy when traversing matter primarily by means of inelastic
interaction with the electrons of atoms in the matter, which excites or ionizes the
atoms. Bethe-Bloch formula is the theoretical description of such energy loss of
charged particle in matter per unit length, as given in 6.1. It depends only on two
properties of the traversing particles, their charge and velocity. It is proportional to
charge square and depends on velocity in a complicated way. Therefore, it provides
a very clean way to distinguish charges of different nuclei by comparing their energy
loss in each velocity bin.
nz 2
dE
47r
dx
mc2
-(
2
2
e2
4Eo
2
)2 [ln(
2mec 2 2
2
-0]
( I (1 - P02)
(6.1)
where 3 = v/c, v is velocity of the traversing particle. z is charge of the traversing
particle. n is electron density of the target. I is mean excitation potential of the
target.
6.1.2
Cluster Selection
Energy deposition on the trackers is measured by the S and K side independently,
so this information is redundant. Because S side has better resolution, only S side is
used for the following analysis.
From the quality control in the preselection, each event has hits reconstructed
from at least 4 tracker layers. The reconstruction program gives a most possible
cluster of energy deposition for the hit on each layer. The energy deposition of an
event can be obtained by averaging the clusters on all layers. There are two categories
of cuts I apply to make sure the clusters used for this calculation are good.
* Examine the energy distribution within the cluster. Each cluster is composed
of at most 5 strips. The strip which has the highest energy deposition should be
the point where the particle hits on, and the strips on its both sides should have
decreasing readings. If there is more than one hump in the energy distribution
among the strips, the cluster has high chance to have been polluted by secondary
particles, thus should be marked as bad cluster. If the highest strip is the
right-most or left-most one, it would probably underestimate the total energy
deposition of the whole cluster. This kind of cluster could be bad, but not
necessarily. I do not simply get rid of the clusters of this kind. This concern is
taken care of in the cuts of the second category.
* Examine the distribution of the clusters composing a single track to get rid of
the outliers. The mean energy deposition is estimated by averaging the clusters
excluding the minimum and the maximum one. The std is approximated by
taking the square root of the estimated mean. Re-examine all the clusters and
keep those within 3o from the estimated mean. The mean energy deposition is
then calculated again by using all the clusters passing the cuts above.
If the incident angle is 0, the actual distance particles traversed would be dx/ cos 0.
Therefore the recorded energy deposition needs to be reduced by cos 0 to obtain the
true energy deposition.
dE
dE
dE
Icos 01
dx
true
dx
(6.2)
record
The energy deposition of all data is plotted in each beta bin, from 0.67 to 1, in
steps of 0.01. Several examples are shown in Figure 6-1, Figure 6-2 and Figure 6-3
for data, boron Monte Carlo and carbon Monte Carlo, respectively.
6.2
Event Selection
6.2.1
Charge Selection
Energy deposition plots in each velocity bin for data and Monte Carlo (MC) are
generated in the same way. A discrepancy between data and MC in the cluster
data, beta=0.85
data, beta=0.79
16
14
12
0o
86
4
2
0
500 1000 1500 2000 2500 3000 3500 4000
Tracker Energy Deposition
data, beta=0.95
data, beta=0.99
!0
5
0
5
0
Tracker Energy Deposition
M LL
I
I 1i....
.. .A.
500 1000 1500 2000 2500 3000 3500 4000
Tracker Energy Deposition
Figure 6-1: Energy deposition on trackers for data in several beta bins. The colored
lines are Gaussian fits.
Bmc, beta=0.85
Bmc, beta=0.79
0-
0
0-
0
00
.00-
0-
0
0
500 1000 1500 2000 2500 3000 3500
401
Tracker Energy Deposition
500 1000 1500 2000 2500 3000 3500 4000
Tracker Energy Deposition
Bmc, beta=0.95
Bmc, beta=0.99
0
010-
Io
i
10
-
100
0
500
1000 1500 2000 2500 3000 3500 4000
Tracker Energy Deposition
500
1000 1500 2000 2500 3000 3500 4000
Tracker Energy Deposition
Figure 6-2: Energy deposition on trackers for boron Monte Carlo in several beta bins.
The colored lines are Gaussian fits.
44
Cmc, beta=0.85
Cmc, beta=0.79
JU
i0
10
;0
tO
!0
o
10
0
0
0
500 1000 1500 2000 2500 3000 3500 4000
Tracker Energy Deposition
Cmc, beta=0.95
500 1000 1500 2000 2500 3000 3500 4000
Tracker Energy Deposition
500 1000 1500 2000 2500 3000 3500 4C
Tracker Energy Deposition
Cmc, beta=0.99
1000 1500 2000 2500 3000
Tracker Energy Deposition
Figure 6-3: Energy deposition on trackers for carbon Monte Carlo for several beta
bins. The colored lines are Gaussian fits.
carbon data
32.5
-
carbon MC
-
2
1.5-
1
I
.o
.
......
.
.
.
0-0.5
-1I
0.1
I
I 11111
I
1
I
I
111111
1
111111
10
Momentum (GeV/nucleon)
100
I
I
11111
1000
Figure 6-4: Comparison of two ratios: circle: the ratio of la data to 2a data; square:
ratio of la MC to 2a MC.
position and width makes it difficult to determine comparable selection criteria for
data and MC. Fortunately, clusters of each element for data and MC have similar
shapes and can be fitted fairly well by a Gaussian. Therefore, my procedure to select
boron and carbon is the following: Fit the data energy deposition histogram with
three Gaussians in a rough region for boron, carbon and nitrogen. For each element,
select events with energy deposition within 2 standard deviations from the mean.
Fitting results for several velocity bins are shown in Figure 6-1, 6-2 and 6-3. It can
be seen that each element separates from each other reasonably well and no significant
contaminations from the neighbor elements are seen. The same Gaussian fit is applied
to boron and carbon MC and the same selection criteria are employed.
Without beam tests to do better fine-tuning of the MC, this "k-sigma" method
is a reasonable way to do the selection with the information handy. A test is done
to verify the feasibility of this method. Data and MC are selected within la and
2a respectively. Taking carbon as an example, the two data spectra are called la
data and 2a data, and the two acceptance matrices are called la acceptance and 2a
acceptance. I then pick a carbon spectrum generated by GALPROP from an arbitrary
model and convolve it with la and 2a acceptance respectively. This produces two
carbon spectra on the detector, which characterize the difference in MC due to 1
or 2a selection criteria. These two spectra were denoted as la MC and 2a MC. If
the "k-sigma" selection criterion is ideally comparable for data and Monte Carlo, the
ratio of la data to 2a data and the ratio of la MC to 2a MC should be exactly the
same. They are plotted in Figure 6-4. They are consistent with each other within
the errors.
There are 25,079 events past 2o selection identified as carbons, and 9,847 events
identified as borons. Their acceptances obtained by similar selection on MC are shown
in Figure 6-7.
6.2.2
Cosmic Origin Selection
The final step of selection is to differentiate borons and carbons of cosmic origin from
those trapped by the Earth magnetic field or produced in the atmosphere (the latter
two categories are known as atmospheric secondary particles). A set of programs [3]
is adopted to trace charged particles in the geomagnetic field. Tracing backwards a
certain amount of time (- 20 seconds), if particles come from sufficiently far away
from the Earth (> 10RD), they are identified as of cosmic origin. The advantage of
using this technique than simply cutting all events with energy below the geomagnetic
cutoff is that more events we can possibly keep for analysis. There are approximately
twice as many events survived by using this method. However, the drawback is that it
is difficult to calculate the exposure time in this case, because there is some probability
that charged particles with energy below the geomagnetic cutoff bouncing off by the
magnetic field, which we do not have much handle on. Fortunately, the quantity we
are interested in is the ratio of borons to carbons, not their absolute spectra. The
exposure time is only a function of rigidity which can be canceled out by taking the
ratio of the two elements. Therefore, we use this tracing technique for selection to
get more statistics.
There are 21,674 carbons and 8,582 borons identified as of cosmic origin (Figure
6-5). There are 3,405 carbons and 1,265 borons identified as atmospheric secondaries
(Figure 6-6).
6.3
Fitting
GALPROP [62] predicts the B/C spectrum given the propagation model and the injection spectra. As discussed before, the diffusive reacceleration model is employed.
B/C has little dependence on the injection spectra. Figure 6-8 shows the comparison
of the effect of varying injection spectra and propagation parameters on B/C. Therefore, in the B/C fit, the injection index is fixed as the default value 2.36 suggested by
Strong and Moskalenko. Only the propagation parameters, which are diffusion coefficient D,, and Alfven velocity VA, are allowed to vary. The fitting method is described
in Appendix B. Boron and carbon spectra generated by GALPROP, modulated by
solar modulation, and convolved with their acceptance matrices respectively, are fit to
4.8, and VA f 30, with x 2 /N = 1.3(N = 18)
our B/C data. The best fit yields D,
(Figure 6-9).
Figure 6-10 shows the X 2 of the fit on the propagation parameter space. D,, and
vA have a strong correlation. The reason is that the poor energy resolution washes
out the difference between the features of peak position and overall height, which are
Cosmic Borons
o 10
102
10
1
0.1
1
10
100
Momentum (GeV/c) per nucleon
1000
Cosmic Carbons
10 3
102
10
1
0.1
1
10
100
1000
Momentum (GeV/c) per nucleon
Figure 6-5: AMSO1 cosmic boron (panel (a)) and carbon (panel (b) counts as a
function of momentum on the detector, before accounting for the acceptance.
Atmospheric Secondary Borons
102
10
1
10
100
1000
Momentum (GeV/c) per nucleon
Atmospheric Secondary Carbons
.I 103
102
10
1
0.1
1
10
100
1000
Momentum (GeV/c) per nucleon
Figure 6-6: AMSO1 atmospheric secondary boron (panel (a)) and carbon (panel (b)
counts as a function of momentum on the detector, before accounting for the acceptance.
Acceptance for boron
1000
).06
).05
U
( 100
U
E
).04
U-
a
E
).03
o 10
).02
I-
i= ia)
010
' ' '""'
.1
1000
).01
1
10
100
Reconstructed Momentum (GeV)
I I I II
1000
Acceptance for carbon
-
0.04
.035
0o
o0
~100 -
m
0.03
UUI
0.025
c 10
0.02
E
a)
0.015
00
1
-_
0.01
b)
0.005
.1
0.. , , ,,, I
I
'
'
'"
"
I
. 0, , ,,,, I
I
.'
.'
'I" ..
...
, ,
, ,
, ,,
.
1
10
100
Recorded Momentum (GeV/nucleon)
1000
Figure 6-7: Acceptance of boron (panel (a)) and carbon (panel (b)) generated by
Monte Carlo.
S
0.3
-
@
different injection, same propagation parameters
- ....different propagation parameters, same injection
S0.25
.........
different injection, different propagation parameter.,-,-;.'
0.2 _-.. .,,
' '
,-.J,.
L
--
n..
S0.05 0
c
-0.050.1
0.1
C
1
100
10
1000
Momentum (GeV/c)
Figure 6-8: Difference of B/C for different models, simulated by GALPROP. Solid
line: with only the injection changed, the difference in B/C is small; dashed line:
with only the propagation parameters changed, the difference is sizable; dotted line:
with both changed, the difference is dominant by the effect from propagation.
the typical information carried by VA and D,,
found for D,,
respectively. A confidence region is
and vA, as described in Appendix B (Figure 6-11).
Alternatively, we plot the x 2 of the fit on D,, vs VA/D space, shown as Figure
6-12, which is attempted to decorrelate the propagation parameters. The effect of
decorrelation by doing this is not significant since the trend line does not go through
the origin.
6.4
Combining With Proton Fit
The galactic propagation model we are seeking is a model which can explain all the
observations self-consistently. Protons are the majority constituent of cosmic rays, so
their measurement is most reliable. The proton spectrum is thus used to investigate
self-consistency of the propagation model. In the proton fit, propagation parameters,
VA and D,,, are fixed, and only the injection index and normalization are allowed
to vary. In Figure 6-13, I survey the proton fit on the propagation parameter space.
A poor proton fit means that no matter how one adjusts the injection, a good fit
can not be found with these propagation parameters. For example, our proton data
disfavor the combinations of low D,, and low VA while the B/C loses the resolving
B/C ratio best fit
1
100
10
Momentum (GeV/c) per nucleon
B/C ration best fit residuals
++
I
I I
II
I
I
I
I I I III
I
10
Momentum (GeV/c) per nucleon
I
I
I
I
I
I
I
100
Figure 6-9: Upper panel: Best fit of B/C ratio on the detector; lower panel: residuals
of the fit.
B/C ratio
u,
E
>
102
4
4.5
5
5.5
6
D, (e28 cm2 s-l)
6.5
7
7.5
Figure 6-10: B/C fit on the propagation parameter space: D,,
vs VA.
B/C ratio 50%, 75% and 90% contour
45-
I
I
I
I
I
I
I
I
i
I
I
I
4.5
I
I
I
I
I
I
.
.i
.
.
II1.
i
.
II
5.5
6
Dxx (e28 cm2 s-1)
Figure 6-11: Confidence regions of B/C fit for Dxx and VA.
xx
20
18
16
14
B/C ratio
10
12
3
102
10
8
6
4
2
10
5
5.5
5
D,, (e28 cm2 s-l)
Figure 6-12: B/C fit on the propagation parameter space: Dxx vs
VA/Dxx.
power along that band. In conclusion, by adding the information from proton fit, our
data favor the right side of the confidence region of Dxx and VA.
6.5
Comparison with Other Measurements
In order to make a comparison of our data with other measurements, we need to
obtain the B/C above the detector. This unfolding is not a trivial problem, since the
acceptance matrix is singular. If we use Singular Value Decomposition class to invert
the matrix directly, a lot of negative numbers are generated in the inverted matrix
which have no physical meaning. Instead, the method of convergent weights [35] [43],
an iterative procedure, is used to solve this problem (see Appendix D).
The unfolded B/C spectrum is shown in Figure 6-14. HEAO3 [29] and Voyager2
[64] data are also plotted in the same figure. Our result is consistent with these
measurements, although ours have larger error bars. Limited statistics above and
below a few GeV are limited by statistical error. Around several GeV, MC systematics
dominate.(see Chpater 9).
6.6
Global Constraints on Propagation Models
Better constraints on propagation models can be obtained by including HEAO3[29]
and Voyager2 [64] data in our fit. Following exactly the same method as fitting
B/C ratio 50%, 75% and 90% contour
4
4.2
4.4
4.6
4.8
5
5.2
5.4
5.6
5.8
D,, (e28 cm2 s-1)
Figure 6-13: Self-consistency check by proton fit. The number is the chi square of the
best proton fit for that particular combination of propagation parameters. The red
number indicates the default set of propagation parameters in GALPROP. The color
code underneath is the 50%, 70% and 90% contour from B/C fit (copied of Figure
6-10).
unfolded B/C ratio at Earth
-- AMSO1
-HEAO3
o 0.4
-
00.35
Vovacer2
0.3
0.25
0.2
0.15
0.1
0.05
0
1
10
momentum (GeV/nucleon)
100
Figure 6-14: Boron to carbon ratio at Earth
5.7, and vA - 35.6, with X2 - 2.2
AMSO1 data only, the best fit yields D,,,
(N = 36). Figure 6-15 is a scan of the fit over the propagation parameter space.
A strong correlation is still present between D and vA. Figure 6-16 indicates the
confidence regions for these two parameters.
chisq of B/C fit on propagation parameter space
104
10,
102
4.5
5
6
5.5
Dxx (e28 cm2 s-1)
6.5
7
Figure 6-15: B/C fit combining AMS01, HEAO3 and Voyager2 data.
B/C ratio fit 75%, 99% contour
5.4
5.5
5.6
5.7
5.8
6
5.9
Dxx (e28 cm2 s-l)
Figure 6-16: Confidence regions of the B/C fit combing AMS01, HEAO3 and Voyager2
data.
58
Chapter 7
Results
7.1
Constraining the Boost Factor
We fit a model of e- backgrounds and x annihilation to e- and p momentum spectra
in order to determine the maximum allowable boost factor consistent with AMS01
Z=-1 data. In the fit, the background normalization, injection index and boost factor
are free parameters. The boost factor is constrained to be non-negative in the fit.
The fit actually relies on the spectral shape - DM annihilation produces spectral
feature on top of an otherwise smooth power-law like spectrum. The best fit of the
boost factors for all SUSY benchmarks [55] are not statistically significantly different
from zero, which means our fitting results are consistent with no dark matter for all
SUSY scenarios with boost factor of unity. However, an upper limit of the boost
factor with certain confidence allowed by our data can be determined. Tables 7.1 and
7.2 show the best fits and the 90% upper bounds of the boost factor, obtained with a
particular set of propagation parameters in diffusive reacceleration (DR) model (the
default values in GALRPOP: D,, = 5.75 x 1028 cm 2 S-1 , vA = 36 km s-l).
A best fit boost factor equal to zero means that no improvement is made by
including the neutralino annihilation products (electrons and antiprotons) for that
particular SUSY scenario. For the benchmark M, the best fit of the boost factor
is a value which could be realized in our Galaxy, while the x 2 has a sizable change
compared with that of the background-only case. Therefore, we would most possibly
obtain a useful upper bound for this scenario. However, for the benchmark H, the
best fit boost factor is too large to be realistic, and the X2 improves even less than
0.5% over the background only case. This analysis is thus insensitive to this particular
scenario. For the benchmark J, even though the best fit boost factor might still be
possible given a very clumpy dark halo, the X2 changes by less than the convergence
criteria of the fitting algorithm. We therefore expect to obtain a very loose upper
bound. in Table 7.2, we see this is the case.
SUSY
Benchmark
A
B
C
D
E
F
G
H
I
J
K
L
M
Boost Factor
lu±
0
0
0
0
0
0
0
26349
0
866
184.2
0
26
17874
412
6863
56261
93
3603
900
86639
130
11436
349
142
44
Background
Normalization
0.729
0.719
0.729
0.729
0.729
0.729
0.729
0.690
0.729
0.710
0.664
0.729
0.658
Background
Index
2.695
2.695
2.695
2.695
2.695
2.695
2.695
2.720
2.695
2.707
2.739
2.695
2.744
Table 7.1: Best fits for all benchmark points.
SUSY
Benchmark
A
B
C
D
E
F
G
H
I
J
K
L
M
90% Upper Bound
of the Boost Factor
52001
1871
16935
77566
364
3731
2811
206454
435
13626
902
262
116
Table 7.2: 90% upper bounds of the boost factor.
x 2 /dof
(dof=13)
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.93
0.95
0.95
0.91
0.95
0.91
7.2
Propagation Model Uncertainties
An optimal region of propagation parameters for the diffusive reacceleration (DR)
model can be deduced by the B/C ratio, which is done in the previous chapter.
AMSO1 data alone produce weak constraint. Combined with the constraints from
proton flux, I define benchmarks on the DR propagation model parameter space, as
shown in Figure 7-1. The default settings of GALPROP is denoted as "S&M", where
Dxx = 5.75 x 1028 Cm2 S-1, VA = 36 km s-1 . I then perform the same fitting as described
in Section 7.1, scanning the propagation parameter space following these benchmark
points. The extracted upper bounds of the boost factor are given in Table 7.3. For
the simplicity for comparison, they are given in terms of the percentage difference
compared to that of the S&M benchmark point. For example, for SUSY model A,
the boost factor 90% upper bound is decreased by 8% using DR model at benchmark
I than that of DR model at benchmark S&M. Under this loose constraint, the upper
bounds of the boost factor change less than 30% resulting from the uncertainty of
modeling the galactic propagation.
The blue line is a linear fit through the reference point S&M and the two points
where the changes in boost factors are smallest, II and VII. We see that above the
line, the changes in the boost factor are negative, while below the line, the changes are
positive. The further away from the line, the bigger the change is. The reason is that
reacceleration produces bumps, so relatively larger Alfven velocity (reacceleration)
leaves smaller room for adding in DM signal, and thus decreases the upper limit
of the boost factor. Therefore, it is actually the spread in the Alfven velocity that
determines the induced uncertainty of the boost factor upper bound. With AMSO1
data alone, the spread is - 5 km/sec.
Much tighter constraints on the propagation parameters can be obtained by adding
HEAO3 and Voyager2 data in the fit. Benchmark points I and II defined in Figure
7-1 are at the lower and upper edge of the 99% confidence contour(Figure 6-11). In
this case, the spread in vA shrinks to ~ 2 km/sec, thus the resultant uncertainty in
the upper bound of the boost factor is down to - 10%.
B/C ratio 50%, 75% and 90% contour
II
15
1
I
4
,
,
,
,
I
4.5
I
5
,
I
I
.
.
I
6
5.5
Dxx (e28 cm2 s-1)
Figure 7-1: Benchmark points on the DR propagation model parameter space, suggested by AMSO1 B/C ratio and proton flux.
SUSY
Benchmark
A
B
C
D
E
F
G
H
I
J
K
L
M
90% Upper Bounds of the Boost Factor
Galactic Propagation Benchmark
VI
V
IV
III
II
I
S&M
4% -21% -30% -11% -23%
-8%
52001
-4% 0% -14% -24% -15% -23%
1871
-8% 4% -20% -30% -11% -22%
16935
-8% 5% -19% -26% -51% -17%
77566
-7% 3% -23% -28% -11% -23%
364
-9% 5% -20% -31% -10% -22%
3731
-8% 3% -21% -29% -12% -22%
2811
206454 -43% 3% -21% -29% -13% -21%
-8% 3% -21% -29% -13% -22%
435
-8% 3% -19% -27% -13% -22%
13626
3% -23% -30% -11% -19%
-7%
902
3% -19% -26% -13% -22%
-6%
262
-7% 3% -18% -26% -10% -18%
116
VII
-1%
-10%
-2%
6%
-2%
-1%
-3%
-5%
-4%
-4%
-4%
-3%
-3%
Table 7.3: 90% upper bounds of the boost factor, scanning on the optimal propagation
parameter region constrained by AMS01 data only. The percentages shown in the
table are the change of the 90% upper bound of the boost factor of that particular
galactic propagation benchmark compared with that of the S&M benchmark.
Chapter 8
PAMELA Results
PAMELA recently released their first results of antiproton and positron [4] [?]. In
this chapter, I investigate the implications of these results on Boost Factor (BF)
constraints.
8.1
8.1.1
The p/p Ratio
Background Only
Figure 8-1 shows the PAMELA p/p data and the GALPROP predictions of our favored propagation parameters assuming no dark matter. The antiproton data can
be explained by the predictions of secondary antiprotons, and are consistent with no
dark matter.
8.1.2
Background + DM Signal
We can add antiprotons from neutralino annihilations for different SUSY benchmarks.
Maximum boost factors can be derived by requiring that the resulting antiproton flux
does not exceed 2a error bars of the PAMELA data in each kinetic energy bin above 10
GeV. Smaller energy bins are not considered because of two reasons: (1) they are not
the sensitive energy region for the DM signal; (2) they suffer from solar modulations.
The results are shown in Figure 8-2. For example, the boost factors can not exceed
120, 21, 43, and 5, for SUSY benchmarks B, E, K and M, respectively. The effect
of propagation model parameters on the BF upper limits is less than an order of
magnitudes in all cases. (see Figure 8-3).
8.2
8.2.1
The e+/(e + + e-) ratio
Background Only
Figure 8-4 shows the PAMELA e+/(e + e-) data and GALPROP predictions of our
favored propagation models assuming no dark matter. The agreements are poor. The
aniproton/proton
0.25
0.2
0.15
0.05
1000
10
Momentum (GeV/c)
Figure 8-1: Simulations of our favored galactic propagation models, assuming no dark
matter. PAMELA antiproton data are from [4].
Allowed boost factors by PAMELA antiproton data
10s
D
10
* 10.1 GeV
* 12.3 GeV
H
A
4
C
E
G!
I
,1E
j
e 15.3 GeV
* 19.5 GeV
1
25.9 GeV
37.3 GeV
61.2 GeV
F
I
L
i
M
-
i
10
Neutralino Mass (GeV)
103
Figure 8-2: Boost factor upper limits allowed by PAMELA antiproton data in each
kinetic energy bin above 10 GeV, for SUSY benchmarks A - M.
Maximum boost factors allowed by PAMELA antiproton data (kinE>10GeV)
with different Galactic propagation parameters
* S&M
5
, 10
s0
D
H
IA
104
C
ill
IV
j
3
10 -
G
G.
B
102
10_=-
- L
F
F
l
*
1 F
Neutralino Mass (GeV)
Figure 8-3: Maximum boost factor upper limits in all kinetic energy bin above 10
GeV, for SUSY benchmarks A - M, using different Galactic propagation parameters.
secondary positrons generated by our galactic propagation models have big excess in
low energy. One argument is that the measurement at low energy is not very reliable
because of the large uncertainties from solar modulation. Figure 8-5 shows the effect
from solar modulation on the e+/(e+ + e-) ratio. It seems that solar modulation
alone can not explain the big discrepancy between the prediction and the data. Some
adjustments in the propagation model are likely necessary. Reducing reacceleration
or adding advection could reduce this discrepancy.
Figures 8-6 and Figure 8-7 show that reducing reacceleration or adding advection
can significantly reduce the excess in low energy region, but the high energy region is
still poorly fit. However, in Chapter 6 we have seen that B/C and proton data disfavor
DR propagation model with very small Alfven velocity. In the next section, we use
the spectra of p/p, B/C and protons to check the consistency of adding advection.
8.2.2
Consistency Check
Adding advection can produce good fit to PAMELA antiproton data (see Figure 8-8).
It is still possible to produce good fit to the B/C spectrum when adding moderate
advection. However, when the advection is too large, the agreement deteriorates.
dvdz = 20 km s-1 kpc-', which is needed to fit PAMELA e+/(e + + e-) data well, is
apparently too large for B/C (See Figure 8-9).
It is similar to the situation of B/C, where moderate advection is acceptable, while
positron/(positron+electron)
-
S&M
x IV
VI
P" :+
10
-
e
.. x,
0 PAMELA
10-2
0.1
1
10
Momentum (GeV/c)
100
1000
Figure 8-4: Simulations of our favored galactic propagation models, assuming no dark
matter. PAMELA positron data are from [?].
too large advection would degrade the fit (See Figure 8-10).
8.2.3
Background + DM signal
There is significant excess of e+/(e+ +-e-) events above 10 GeV/c. Positron flux from
neutralino annihilations could be one possible explanation. However, large boost
factors are required to give a reasonable fit (see Figure 8-11). If the excess results
from neutralino annihilation, it should have a cut-off somewhere before the neutralino
mass, which should be at several tens to several hundreds GeV. Therefore it would
be extremely important to measure the spectrum of e+l/(e+ + e-) to higher energies
to address the source of this excess.
positron/(positron +electron)
,,,,,,.. )I~ln~llI.I*I
II
ll~r
111
*I1
10-1
S&M, phi=200MV
-
... S&M, phi=800MV
PAMELA
o
10. 2
,, ,,
,,I
, ,, , ,
,,
100
10
Momentum (GeV/c)
1000
Figure 8-5: Effect of solar modulation on e+/(e + + e-). Phi is the parameter in the
forced field approximation of solar modulation, described in Chapter 3.3.
positron/(positron+electron)
10-1
S&M, Va=36
--
... S&M, Va=O
o PAMELA
10 -2
I
I
I
I I.
I
I
,
I
,
,
,,,,
I
'
I
I
I
I
11
100
1000
Momentum (GeV/c)
Figure 8-6: Effect of reducing reacceleration on e+/(e + + e-). Solid line: DR model,
VA = 36 km/s; dotted line: DR model , vA = 0;
positron/(positron+electron)
"r''r
rr
~~
"" "
10-1
-
-
.ri.
0
-
-S&M
..... S&M+ dvdz=l 0
dvdz=20
o PAME LA
10-2
I
1 1111111
1
bM
1 1111111
10
1
) 1111111
100
1
1 111111
1000
Momentum (GeV/c)
Figure 8-7: (Effect of adding advection on e+/(e+ + e-). Solid line: DR model with
parameters at S&M point. Dashed line: DR model at S&M point + advection with
dvdz = 10 km s- 1 kpc-'; dotted line: DR model at S&M point + advection with
dvdz = 20 km s-1 kpc -1
aniproton/proton
x1 01
0.0025
-
-S&M
S - SM+dvdz=10
0.002
....
SM+dvdz=20
O PAMELA
0.0015
"'"~
,,,,.. I
-
-
ri,
,,,,
0.001
0.0005
irill
~,rii
~~,
h,~,
n;n
rn
Z
-
~B
;,
'"""
' '"""
10
' '""'
100
1000
Momentum (GeV/c)
Figure 8-8: Solid line: DR model with parameters at S&M point; dashed line: DR
mode at S&M point + advection with dvdz = 10 km s- 1 kpc-1; dotted line: DR model
at S&M point + advection with dvdz = 20 km s-1 kpc 1 .
B/C with different advection
o 0.45
0.4
-S&M
..... S&M+dvdz=10
.. S&M+dvdz=20
* AMSO1
o HEAO3
F.
0.35
0.3
0.25
0.2
0.15
u~r
I~
-S
A
aI
Voyager2
r
-i$
-
-'
0.1
)I
yl~l
1)~1111
(I ~)~I
l(rllll
(It)
)~
)II
-
0.05
' '"""
' '"""
' ' " ' ''''''''
' '""'
10
100
Momentum (GeV/c) per nucleon
1000
Figure 8-9: Solid line: DR model with parameters at S&M point. Dashed line: DR
mode at S&M point + advection with dvdz = 10 km s- 1 kpc-1; dotted line: DR model
at S&M point + advection with dvdz = 20 km s-1 kpc -1.
Proton
10-2
-
SS&M
---- S&M+dvdz=l 0
-- S&M+dvdz=20
* AMSO1 as fit
A
1-3
10
T1-
-O=.
-t
10-s
10-6 _---0. 1i
.
. . . .
,
, , , , ,I
., .
,
10
Momentum (GeV/c)
......
, ,
,
I
100
,
,
,
, ,,
1000
Figure 8-10: Solid line: DR model with parameters at S&M point. Dashed line: DR
mode at S&M point + advection with dvdz = 10 km s- 1 kpc-1 ; dotted line: DR model
at S&M point + advection with dvdz = 20 km s- 1 kpc -1 . Solid circles: fit from AMS01
proton data.
positron/(positron+electron)
positron/(positron+electron)
-no
-no DM
DM
.... SUSY M, boost=23
,, SUSY D, boost=15000
O PAMELA
OPAMELA
-=
1
10-
10-1
0.1
10.2
"
10.2
1
100
10
Momentum (GeV/c)
0.1
1000
1
100
10
Momentum (GeV/c)
1000
Figure 8-11: Examples of e+/(e+ + e-) with neutralino annihilation for SUSY benchmarks D (left) and M (right). Black line: simulation assuming no dark matter; gray
line: simulation with signal from neutralino annihilation, multiplying by a boost. Use
DR model at S&M point + advection with dvdz = 20 km s-1 kpc-1 for the Galactic
propagation model.
70
Chapter 9
Conclusion
9.1
Main Message of This Analysis
The 90% upper limits on the boost factor for 13 cMSSM benchmark scenarios [17]
are derived, and the effect of CR propagation model uncertainties is assessed. A
diffusive reacceleration model is employed to describe the propagation process. An
optimal parameter region is obtained from the AMS01 measurement of the B/C ratio
and the proton flux. The diffusion coefficient (D,,) and Alfven velocity (VA) are
along a correlated band from lower left to upper right within the region of D,, from
4.5 x 1028 cm 2 s- to 6 x 1028 cm2 s- 1 and VA from 28 km s-1 to 42 km s- 1, as shown in
Figure 7-1. The resultant uncertainty on the 90% upper bounds of the boost factor
is shown to be less than 30%, except for SUSY D, DR V (-51%), and SUSY H, DR I
(-43%). Better constraints on the propagation parameters are deduced by including
more accurate measurements of the B/C ratio (HEAO3 [29] and Voyager2 [64]), which
shrinks the induced uncertainty on the upper bounds of the boost factor to - 10%.
In conclusion, the resultant error on dark matter limits in the Z = -1 spectrum from
the uncertainty of galactic propagation modeling is not significant. However, because
propagation models can mimic neutralino annihilation in some SUSY models, the
impact of propagation models must be studied.
9.2
Broaden the Interpretation of the Boost Factor
Since some of the boost factor limits we get are very large, a natural question to ask
would be what is a reasonable number. A recent study showed that the boost factor
associated to DM clumps could not exceed at most a factor of - 10 in the standard
A-CDM scenario of structure formation [46]. However, the annihilation rate not only
depends on the p2m, but also depends on the annihilation cross-section. The crosssection could be enhanced by the Sommerfeld effect [45], which is a quantum effect,
arising because, in the interaction the wave function of the particles is distorted by
the presence of a potential if their kinetic energy is low enough. DM is very non-
relativistic, so they could have the Sommerfeld effect. Such enhancement S goes
as 1/3, S - wa// [45], where a is the fine structure constant. Take / = 10- 3 , we
estimate S ~ 23. In the clumps, DM particles would have even smaller kinetic energy,
therefore a larger Sommerfeld enhancement would be expected for them. The boost
factor can thus be interpreted as the product of the clumpiness enhancement and the
Sommerfeld enhancement. 102 to 103 will still be reasonable for a boost factor.
9.3
Message from PAMELA First Results
PAMELA antiproton data is consistent with no DM signal. A maximum boost factor
can be derived by requiring the resultant flux not to exceed 2o7 error bars of the data
(see Figure 8-2). Uncertainties in the Galactic propagation parameters introduce
changes in this upper limit of the boost factor, but it will not affect the order of
magnitude estimates (see Figure 8-3).
PAMELA sees a significant excess in the e+ data above 10 GeV. DM annihilation
could be a plausible explanation, however, it would require large boost factors. Also,
this rising trend is so steep that most of the SUSY benchmark scenarios cannot
produce signals to accommodate it. Measurements to higher energy will help address
the source of this excess. A cut-off at the energy region of several tens to several
hundreds GeV is expected because no charged final states could be produced at
energy higher than the neutralino mass.
PAMELA e+/(e + + e-) data below 10 GeV require little reacceleration or large
advection in the Galactic propagation model. However, such propagation models
contradict the constraints from B/C measurements. Alternatively, solar modulation
is hoped to explain the discrepancy, but the force field model seems to fail to do this
job.
9.4
9.4.1
Future Prospects
B/C Ratio Measurement
Figure 9-1 shows all the constituents from which AMS01 B/C measurement error
budget is built up, where we can get some indications on how to improve this measurement in the future, probably with AMS02. At the momentum of around several
GeV/c per nucleon, where the peak is supposed to be located, the systematic error
resulted from imperfect energy resolution dominates. This would wash out the peak,
which is one of the main reasons for the poor constraints on propagation parameters
placed by our data. AMS02 will have a bending power 5 times stronger than that of
AMSO1 [23], thanks to the strong superconducting magnet, which can improve the
energy resolution. For lower energies (< 1 GeV/nucleon) and higher energies (> 5
GeV/nucleon), data statistical error quickly take over. Having too few events lower
than 1 GeV makes it more difficult to recognize the peak, which is another main
reason for the poor constraints. AMS02 is supposed to take data for three years to
accumulate enough statistics, so this error should be reduced significantly.
Contributions of different components to B/C error
1
data statistics
MC systematic
Mc statistics
0. 8-
0. 6-
0. 4-
0. 2*
n
0.1
10
•
100
1000
Momentum (GeV/c) per nucleon
Figure 9-1: Constituents of the error budget of AMS01 B/C ratio measurement.
9.4.2
Moving to Positron and Antiproton Channels
Positron and antiproton signals from dark matter annihilation have similar strength
as that of electrons, while having very little background from astrophysical sources.
Thus they are much attractive channels to search for dark matter.
The precise measurements of positrons from several GeV to 300 GeV need to reduce the dominating proton background by a factor of 106. This would be achieved
by AMS02, where its electromagnetic calorimeter would deliver 3 to 4 orders of magnitude and its Transition Radiation Detector (TRD) would deliver another factor of
100 to 1000 Figure [40]. The geometric acceptance of AMS02 is about 400 times of
that of PAMELA, which means if run for 3 years, AMS02 could reduce the error bar
of current PAMELA measurements by more than a factor of 25. Such high accuracy
measurements would start shedding light on dark matter.
Appendix A
Derivation of Alfven Velocity
Magnet
d li
Figure A-i: Mechanism of the generation of Alfven waves.
The interstellar matter, which is mainly composed of hydrogen and helium. can
be seen as a perfectly conducting fluid. To balance inertial accelerations and accelerations caused by magnetic field curvature, we have VA/TA= B 2/(LAp), where v is the
fluid velocity, TA is the time scale of the inertial magnetic oscillation, LA is the length
scale associated with the oscillation, B is the galactic magnetic field strength, p is
the interstellar matter density. The conductivity is so high that the magnetic lines of
force are attached to the interstellar matter and partaking in its streaming motions.
Thus we have vA = LA/TA. Combining these relationships, the Alfven velocity is
estimated as VA = Blp1 /2
The following calculation is according to H. Alfven's 1942 paper [8]. Consider the
simple case where the constant magnetic field Bo is homogeneous and parallel to the
z-axis. In order to study a plane wave, we assume all variables depend on time t and z
only. The fluid velocity v is parallel to the x-axis. This motion of the conducting fluid
gives rise to an electromotive force (emf) which produces Electric currents parallel to
the y-axis (See Figure A-1). Assume p = 1.
1
vBo
1_
x Bo = --- c e
c
(A.1)
This produces a variable magnetic field B parallel to the x-axis.
1DB
S=
c dt
D -vBo
()ex
z
c
x E=
E
(A.2)
Multiply A with Equation A.2 gives:
1 D2 B
d 2v
=
-Bo
c2 0 t 2
ataZ
--
(A.3)
The hydrodynamic equation:
Dv
p-- =
1
at
c
(i x Bo)-
Vp
(A.4)
Where the current i parallel to y-axis us generated by:
47.
=V x B=
aB-
(A.5)
Plug Equation A.5 in Equation A.4, we get:
Dv
at
Pt
1 DB
2
zBo
4-FC2 az
4c
=
- Vp
(A.6)
Vx Equation A.6 gives:
D2 v
1 D2 B
4 2 z Bo
47c2 aZ2
(A.7)
B 0 2 D2 B
- =
47rp Dz2
(A.8)
p-azat Combine Equation A.3 and Equation A.7, we get:
a2B
Dt2
Since B has a plane wave solution: B oc ei(wt- kz), thus
B2
0
47 p
=
A2 A =
sqrt4irp
(A.9)
76
Appendix B
GALPROP Parameters (default
setting)
value
Title = conventional/2D 4 kpc tuned to agree with ACE
Title = source isotopic distr. of an element = solar isot. abund. distr.
nspatialdimensions = 2
r_min =00.0
min r
r_max =20.00
max r
dr = 1.0
delta r
min z
z_min =-4.0
max z
zmax =+4.0
dz = 0.1
delta z
min
x_min = 0.0
ax x
x.max =+15.0
dx = 0.2
delt ax ymin = 0.0
nnax y
y_max =+15.0
dy = 0.2
delt ay
p_min =1000
p_max =4000
pfactor =1.50
min y
min momentum (MV)
max momentum
momentum factor
Ekin_min =1.0el
Ekin_max =1.0e7
Ekin_factor =1.2
pEkin_grid = Ekin
E_gammamin = 1. e0
Egammamax = 1.e8
E_gammafactor = 1.4
min kinetic energy per nucleon (MeV)
max kinetic energy per nucleon
kinetic energy per nucleon factor
p-
Ekin alignment
min gamma-ray energy (MeV)
max gamma-ray energy (MeV)
gamma-ray energy factor
integration mode = 1
integr.over part.spec.: =1-old E*logE; 0=1-PL analyt.
nusynchmin = 1 .Oe6
.0e6
min synchrotron frequency (Hz)
max synchrotron frequency (Hz)
synchrotron frequency factor
nu_synchmax = 1.Oe10
nusynchfactor = 2.0
long-min = 0.50
longmax =359.50
latmin =-89.50
lat_max =+89.50
dlong = 1.00
dlat = 1.00
gamma-ray intensity skymap longitude minimum (deg)
gamma-ray intensity skymap longitude maximum (deg)
gamma-ray intensity skymap latitude minimum (deg)
gamma-ray intensity skymap latitude maximum (deg)
gamma-ray intensity skymap longitude binsize (deg)
gamma-ray intensity skymap latitude binsize (deg)
diffusion coefficient at reference rigidity
DO_xx =5.75e28
reference rigidity for diffusion coefficient in MV
Drigid_br =4.0e3
diffusion coefficient index below reference rigidity
D_g_l = 0.34
diffusion coefficient index above reference rigidity
D_g_2 = 0.34
0=no reacc.; 1,2=incl.diff.reacc.; -1==beta3 Dxx; 11=Kolmogorov+damping;
diffreacc =1
12=Kraichnan+damping
Alfven speed in km s-i
v_Alfven =36
MV -some rigidity (where CR density is low)
damping_p0 = 1.e6
a const derived from fitting B/C
damping_const_G = 0.02
Lmax 1 kpc, max free path
damping_max_path_L = 3.e21
convection =0
v0_conv =0.
dvdzconv =5.
1=include convection
v_conv=v0 _conv+dvdz_conv*dz
km s-i
v_conv=v0_conv+dvdz_conv*dz
km s-1 kpc-1
reference rigidity for nucleus injection index in MV
nucrigid_br =9.0e3
nucleus injection index below reference rigidity
nuc_gl =1.82
nucleus injection index index above reference rigidity
nuc_g_2 =2.36
injspectrum_type = rigidity
spectrum type
rigidity-betarig-
-Etot nucleon injection
electron injection index below electronsigid_br0
electron_g_0 =1.60
reference rigidity0 for electron injection index in MV
electron_rigidbr0 =4.0e3
electron injection index below reference rigidity
electron_g_1 =2.5
reference rigidity for electron injection index in MV
electron_rigid_br =1.0e9
electron injection index index above reference rigidity
electron_g_2 =5.0
He/H of ISM, by number
He_Hratio =0.11
X_CO =0.4E20, 0.4E20, 0.6E20, 0.8E20, 1.5E20, 10.0E20, 10.0E20, 10.0E20, 10.0E20
conversion factor from CO integrated temperature to H2 column density
for CO rings 0.0 - 1.5 - 3.5 - 5.5 - 7.5 - 9.5 - 11.5 - 13.5 - 15.5 - 50 kpc
fragmentation =1
1=include fragmentation
momentumlosses =1
l=include momentum losses
radioactive_decay =1
1=include radioactive decay
Kcapture =1
1=include K-capture
start_timestep = 1.0e7
endtimestep =1.0e2
timestepfactor =0.25
timesteprepeat =20
number of repeats per timestep in timetepmode=1
timesteprepeat2 =0
number of timesteps in timetepmode=2
timestepprint =10000
number of timesteps between printings
timestepdiagnostics =10000
number of timesteps between diagnostics
control_diagnostics =0
control detail of diagnostics
networkiterations = 2
propr
propx
prop_y
propz
prop_p
=
=
=
=
=
number of iterations of entire network
1=propagate
1=propagate
1=propagate
1=propagate
1=propagate
usesymmetry = 0
try by copying(3D)
vectorized = 0
in r (2D)
in x (2D,3D)
in y (3D)
in z (3D)
in momentum
0=no symmetry, 1=optimized symmetry, 2=xyz symme-
0=unvectorized code, 1=vectorized code
sourcespecification = 0
2D::1:r,z=0 2:z=0 3D::l:x,y,z=0 2:z=0 3:x=0 4:y=0
source_model = 1
0=zero 1=parameterized 2=Case&B 3=pulsars 4= 5=S&Mattox
6=S&Mattox with cutoff
sourceparameters_l = 0.5
model 1:alpha
model 1:beta
source_parameters_2 = 1.0
source_parameters_3 = 20.0
model 1:rmax
ncr-sources = 0
cr-source x_0l = 10.0
crsourcey_01 = 10.0
cr-sourcez_01 = 0.1
cr-sourcew_01 = 0.1
cr-sourceL_01 = 1.0
cr-source x_02 = 3.0
cr-sourcey_02 = 4.0
cr-sourcez_02 = 0.2
number of pointlike cosmic-ray sources 3D only!
x position of cosmic-ray source 1 (kpc)
y position of cosmic-ray source 1
z position of cosmic-ray source 1
sigma width of cosmic-ray source 1
luminosity of cosmic-ray source 1
x position of cosmic-ray source 2
y position of cosmic-ray source 2
z position of cosmic-ray source 2
crsource_w_02 = 2.4
crsource_L_02 = 2.0
sigma width of cosmic-ray source 2
luminosity of cosmic-ray source 2
handle stochastic SNR events
SNRevents = 0
time interval in years between SNR in 1 kpc-3 volume
1.0e4
SNR_interval=
CR-producing live-time in years of an SNR
SNR_livetime = 1.0e 4
delta electron source index Gaussian sigma
SNRelectron_sdg = 0.00
delta
nucleus source index Gaussian sigma
0.00
=
SNRnuc_sdg
delta electron source index pivot rigidity (MeV)
SNR_electron_dgpivo t = 5.0e3
delta nucleus source index pivot rigidity (MeV)
SNRnuc_dgpivot = 5.0e3
HIsurvey = 9
COsurvey = 9
HI survey 8=orig 8 rings 9=new 9 rings
CO survey 8=orig 8 rings 9=new 9 rings
zzz=10*zscale
rrr=10*rscale
bbb=10*B(0)
bbbrrrzzz
Bfieldmodel = 050100020
=
MilkyWay_DR0.5_DZO.1_DPHI10_RMAX20_ZMAX5_galpropiormat.fits
ISRFfile
input ISRF file
ISRF factors for IC calculation: optical, FIR, CMB
ISRF_factors = 1.0,1.0,1.0
protonnorm_Ekin = 1.00e+5 proton kinetic energy for normalization (MeV)
protonnormflux = 4.90e-9 to renorm nuclei/flux of protons at norm energy (cm-2
sr 1 s 1 MeV-1)
electron_norm_Ekin = 34.5e3
electron_normflux = .40e-9
s l MeV-1)
maxZ = 28
useZ_1 = 1
useZ_2 = 1
useZ_3
useZ_4
useZ_5
useZ_6
=
=
=
=
1
1
1
1
useZ_7 = 1
useZ_8 = 1
use_Z_9 = 1
use_Z_10 = 1
use_Z_11 = 1
use_Z_12 = 1
use_Z_13 = 1
use_Z_14 = 1
use_Z_15 = 1
use_Z_16 = 1
electron kinetic energy for normalization (MeV)
flux of electrons at normalization energy (cm-2 sr-^
maximum number of nucleus Z listed
useZ_17
useZ_18
useZ_19
useZ_20
useZ_22
useZ_23
use_Z_24
use_Z_25
use_Z_26
useZ_27
useZ_28
useZ_29
useZ_30
use_Z_1 = 1
isoabundance01 00 1
isoabundance_01_002
isoabundance_02_003
isoabundance_02_004
isoabundance_03006
isoabundance_03_007
isoabundance_04 009
isoabundance_05010
isoabundance_05 011
isoabundance_06 012
isoabundance_06013
isoabundance_07014
isoabundance_07_015
isoabundance_08016
isoabundance_08_017
isoabundance_08018
isoabundance_09_019
isoabundancel10020
isoabundance_10021
isoabundance_10_022
isoabundance 1_023
isoabundance_12_024
isoabundance_12_025
isoabundance_12_026
isoabundance_13027
isoabundance_14_028
isoabundance_14_029
isoabundance_14_030
isoabundance_15_031
isoabundance_16_032
isoabundance_16_033
1.06e+06
34.8
9.033
He
7.199e+04
0
Li
0
0
Be
0
B
0
2819
C
5.268e-07
182.8
N
5.961e-05
3822
O
6.713e-07
1.286
2.664e-08
Ne
312.5
0.003556
100.1
22.84
Na
658.1
Mg
82.5
104.7
76.42
Al
725.7
Si
35.02
24.68
4.242
P
89.12
S
0.3056
isoabundance_16034 =
isoabundance_16036 =
isoabundance_17035 =
isoabundance_17_037 =
isoabundance_18036 =
isoabundance_18_038 =
isoabundance_18_040 =
isoabundance_19039 =
isoabundance_19040 =
isoabundance_19041 =
isoabundance_20_040 =
isoabundance_20041. =
isoabundance_20042 =
isoabundance_20_043 =
isoabundance_20_044 =
isoabundance_20_048 =
isoabundance_21 045 =
isoabundance_22046 =
isoabundance_22047 =
isoabundance_22_048 =
isoabundance_22049 =
isoabundance_22050 =
isoabundance_23_050 =
isoabundance_23_051 =
isoabundance_24_050 =
isoabundance_24_052 =
isoabundance_24053 =
isoabundance_24054 =
isoabundance_25_053 =
isoabundance_25_055 =
isoabundance_26054 =
isoabundance_26056 =
isoabundance_26 057 =
isoabundance_26058 =
isoabundance_27059 =
isoabundance_28058 =
isoabundance_28_060 =
isoabundance_28061 =
isoabundance_28_062 =
isoabundance_28064 =
3.417
0.0004281
Cl
0.7044
0.001167
Ar
9.829
0.6357
0.001744
K
1.389
3.022
0.0003339
Ca
51.13
1.974
1.134e-06
2.117e-06
9.928e-05
0.1099
Sc
1.635
Ti
5.558
8.947e-06
6.05e-07
5.854e-09
6.083e-07
V
1.818e-05
5.987e-09
Cr
2.873
8.065
0.003014
0.4173
Mn
6.499
1.273
Fe
49.08
697.7
21.67
3.335
Co
2.214
Ni
28.88
11.9
0.5992
1.426
0.3039
total_crosssection = 2
cross_section_option = 012
j=11=Webber, 21=ST
total cross section option: 0=L83 1=WA96 2=BP01
100*i+j i=1: use Heinbach-Simon C,O-,B j=kopt
tlhalflimit = 1.0e4 year - lower limit on radioactive half-life for explicit inclusion
primaryelectrons = 1
secondary_positrons = 1
secondary_electrons = 1
secondaryantiproton = 2
tertiary_antiproton = 1
secondary_protons = 1
012
gamma_rays = 0
1=compute gamma rays, 2=compute HI,H2 skymaps separately
1= old formalism 2=Blattnig et al.
pi0_decay = 0
IC_isotropic = 1
1,2= compute isotropic IC: 1=compute full, 2=store skymap
components
IC_anisotropic = 1
1,2,3= compute anisotropic IC: 1=full, 2=approx., 3=isotropic
bremss = 0
1=compute bremsstrahlung
synchrotron = 0
1=compute synchrotron
comment = the dark matter (DM) switches and user-defined parameters
1=compute DM positrons
DMpositrons = POSITRON_ON
1=compute DM electrons
DMelectrons = ELECTRON_ON
DMantiprotons = ANTIPROTON_ON
1=compute DM antiprotons
1=compute DM gammas
DMgammas = 0
= 2.8
= 0.43
= ENEIRGY
= 40.
= 40.
= 40.
= 30.
= 50.
= 40.
=3.e-25
DMdoubleO
DMdoublel
DM_double2
DM_double3
DM_double4
DM_double5
DM_double6
DM_double7
DM_double8
DM_double9
DM_int0
DMintl
DM_int2
DM_int3
DM_int4
DM_int5
DM_int6
DM_int7
DM_int7
DM_int9
=1
=1
= 1
=1
= 1
=1
= 1
=1
=1
= 1
core radius, kpc
local DM mass density, GeV cm-3
neutralino mass, GeV
positron width distribution, GeV
positron branching
electron width distribution, GeV
electron branching
pbar width distribution, GeV
pbar branching
icrosssec *VL-thermally overaged, cm3 s-1
isothermal profile
output_gcrfull = 0
warmstart = 0
verbose = 0
test_suite = 0
output full galactic cosmic ray array
read in nuclei file and continue run
verbosity: O=min,10=max iO: selected debugs
run test suite instead of normal run
Appendix C
Unfolding Method: Convergent
Weights
What we measure on the detector can be expressed as
f (p) --
A(p, p')g(p')dp'
(C.1)
where f(p) is the recorded spectrum, g(p') is the incident spectrum, A(p, p') is the
acceptance matrix, which is the probability that a particle with momentum p' is
detected with momentum p. f(p) is known and we want to invert the acceptance
matrix to obtain the incident spectrum g(p'). This is not a trivial problem, since the
acceptance matrix is singular. If we use Singular Value Decomposition class to invert
the matrix directly, a lot of negative numbers are generated in the inverted matrix
which have no physical meaning. Instead, the method of convergent weights [35] [43]
is used to solve this unfolding problem. The iterative procedure is formulated as
following:
(C.2)
gJ+1 (p,) - fr (p)A(p, p')dp
A(p, p')dp
where
f(p)
r(p) =
(P)
(C.3)
f A(p, p') gi(p')dp'
There are two main sources of errors, one is the statistics error from data, the other is
the systematic error from the acceptance matrix (the statistic error from acceptance
can be reduced to an ignorable level by generating more Monte Carlo events). The
errors are propagated quadratically.
86
Appendix D
FitMethod
X2 minimization (Equation D.1 or log likelihood maximization ?? is used in the fitting.
Gaussion error is assumed in the fit, so the two methods are accentually equivalent.
(d,
fhiSquare
IC
-
yi)2
_
iebins
(d - yi)2
fLogLikelihood
((d_-i)2
-
zEbzns
(D.1)
Z
(D.2)
i
where index i stands for the momentum bin. di and yi are data and theoretical
prediction at momentum bin i respectively. a is the total error including statistical
error from data and systematic error encoding in the acceptance, a? = istat + , sys"
The program MINUIT [2]is used to do this optimization problem and a gradient
descent method is chosen.Uncertainty for the fit parameter a is calculated as a, =
d2f)-1
The difference between the value of the fit function (X2 or log likelihood) and that
of the best-fit point determines the contours of equal likelihood, see Table D-1. There
are three free parameters in the DM boost factor fit, so we use AX 2 = 6.25 to set
the contour of 90% confidence ellipsoid. There are two free parameters in the B/C
ratio fit, so we use AX2 = 1.39, 2.41, and 4.61 to set the 50%, 70% and 90% confidence
ellipsoid respectively.
Number of
Confidence level (probability contents desired inside
+ UP)
hlypercontour of k = X ,in
50%
70%
90%
95%
99%
0.46
1.07
2.70
3.84
6.63
S1.39
2.41
4.01
5.99
9.21
3
2.37
3.67
6.25
7.82
11.36
4
3.36
4.88
7.78
9.49
13.28
5
4.35
6.06
9.24
11.07
15.09
6
5.35
7.23
10.65
12.59
16.81
7
6.35
8.38
12.02
14.07
18.49
8
7.34
9.52
13.36
15.51
20.09
9
8.34
10.60
14.68
16.92
21.67
10
9.34
11.78
15.99
18.31
23.21
11
10.34
Parameters
1
24.71
19.08
17.29
If FCN is - log(likelihood) instead of X 2, all values of UP
should be divided by 2.
12.88
Figure D-1: Table of UP for multi-parameter confidence regions [2]
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